Descriptions

Figure 1 Stemplot of coffee prices from Table 1

An ordered stemplot of coffee prices from Table 1. The first column contains the numbers in the stem while the second column contains the relevant leaf values. There are 11 levels. The numbers in the stem start at 26, increase in steps of one and end at 36. The leaf values correspond to the relevant numbers in the stem, though sometimes there is more than one leaf and sometimes none at all. In this stemplot they are in numerical order.

At level 26 there are five leaves, 8, 8, 8, 8, 9. Level 27 has two leaves, 5, 9. Level 28 has no leaves while level 29 has five leaves, 5, 5, 5, 5, 9. Level 30 has a single leaf, 5, and level 31 has a single leaf, 5. Levels 32, 33, 34 and 35 have no leaves. Level 36 has a single leaf, 9. Beneath the stemplot is written n = 15, followed by 26 vertical line 8 represents 268 pence. The vertical line sits horizontally between the 26 and the 8.

Back to - Figure 1 Stemplot of coffee prices from Table 1


Figure 2 Subscript notation for ordered data

Subscript notation for ordered data. This shows a table with 2 rows and 15 columns. The first row contains x subscript 1, x subscript 2 and so on, ending at x subscript 15. The subscript numbers are in round brackets and positioned to the right of and slightly below each x. Above this row is a cloud with an arrow pointing to the term x subscript 3. In the cloud is written ‘The subscript is (3), so this is the third value in the ordered batch.’

The second row contains the corresponding values. These are 268 (shown below x subscript 1), 268, 268, 268, 269, 275, 279, 295, 295, 295, 295, 299, 305, 315 and 369 (shown below x subscript 15). Below the table are three clouds with arrows pointing upwards. The first cloud contains ‘capital E subscript capital L’ and the arrow points to the first number in the second row, 268. The middle cloud contains the word ‘Median’ and the arrow points to the first 295 in the second row, which is under x subscript 8. The last cloud contains ‘capital E subscript capital U’ and the arrow points to the last number in the second row, 369.

Back to - Figure 2 Subscript notation for ordered data


Figure 3 Median of 15 values

Median of 15 values. The letters x subscript 1, x subscript 2 and so on, each with the subscripts in round brackets, are arranged in order, but in a V-shaped formation. At the top left-hand end of the V is x subscript 1. The letters then descend in order down the slope until they reach x subscript 8. Then they begin to rise up a slope to form the right-hand side of the V, starting with x subscript 9, which is level with x subscript 7. They continue upwards to end at x subscript 15, which is level with and to the far right of x subscript 1. Below the point of the V is a cloud containing the word ‘Median’. From the cloud an arrow points to x subscript 8.

Back to - Figure 3 Median of 15 values


Figure 4 Prices of 10 digital cameras

V-shaped formation for prices of ten digital cameras. Reading down the slope from the top left, the numbers are 53, 60, 65, 70 and 70, but this V shape has no number at its point. The number 74 is to the right of but level with the preceding number 70. The remaining numbers 79, 81, 85, 90 then rise up the slope, with 90 being level with and to the far right of 53.

Back to - Figure 4 Prices of 10 digital cameras


Figure 5 Prices of all flat-screen televisions with a screen size of 24 inches or less on a major UK retailer’s website on a day in February 2012

Stemplot with 10 levels, though there are repeated numbers in the stem. The numbers in the stem are 0, 1, 1, 1, 1, 1, 2, 2, 2, 2. At level 0 there is one leaf 9. At the first level 1 there is one leaf, 0. At the second level 1 there are four leaves 2, 3, 3, 3. The third level 1 has five leaves, 4, 5, 5, 5, 5. The fourth level 1 has three leaves, 6, 6, 7. The fifth and last level 1 has three leaves, 8, 8, 9. The first and second level 2s have no leaves. The third level 2 has two leaves, 4, 5. The last level 2 has one leaf, 7. Beneath the stemplot is written n = 20, followed by 0 vertical line 9 represents 90 pounds sterling. The vertical line sits horizontally between the 0 and the 9.

Back to - Figure 5 Prices of all flat-screen televisions with a screen size of 24 inches or less on a major UK retailer’s website on a day in February 2012


Figure 6 Stemplot of 14 gas prices

A stemplot with 8 levels. The numbers in the stem start at 374, increase in steps of one and end at 381. At level 374 there are 3 leaves, 0, 0, 3. Level 375 has no leaves. Level 376 has two leaves, 0, 7 while level 377 has one leaf, 6. Level 378 has one leaf, 4. Level 379 has two leaves, 5, 6. Level 380 has four leaves, 1, 1, 4, 5. Level 381 has one leaf, 8. Beneath the stemplot is written n = 14, followed by 374 vertical line 0 represents 3.740 pence per kilowatt hour. The vertical line sits horizontally between the 374 and the 0.

Back to - Figure 6 Stemplot of 14 gas prices


Figure 7 Stemplots for northern and southern cities separately.

There are two stemplots, one for northern cities and one for southern cities. The one for the northern cities has 7 levels. The numbers in the stem start at 374, increase in steps of one and end at 380.

At level 374 there are 2 leaves, 0, 0. Level 375 has no leaves. Level 376 has one leaf, 7 and level 377 also has one leaf, 6. Levels 378 and level 379 have no leaves. Level 380 has three leaves, 1, 1, 4. Beneath the stemplot is written n = 7, followed by 374 vertical line 0 represents 3.740 pence per kilowatt hour. The vertical line sits horizontally between the 374 and the 0. The one for the southern cities has 8 levels. The numbers in the stem start at 374, increase in steps of one and end at 381. At level 374 there is one leaf, 3. Level 375 has no leaves. Level 376 has one leaf, 0 but level 377 has no leaves. Level 378 has one leaf, 4, and level 379 has two leaves, 5, 6. Level 380 has one leaf, 5 and level 381 has one leaf, 8.

Beneath the stemplot is written n = 7, followed by 374 vertical line 3 represents 3.743 pence per kilowatt hour. The vertical line sits horizontally between the 374 and the 3.

Back to - Figure 7 Stemplots for northern and southern cities separately.


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3.818 minus 3.740=0.078 comma

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3.804 minus 3.740=0.064 comma

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3.818 minus 3.743=0.075.

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mean = fraction sum over size end .

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fraction sum over size end = fraction 4 + 8 + 4 + 2 + 9 over 5 end = fraction 27 over 5 end = 5.4.

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mean = fraction sum over size end

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overline x = fraction sum x over n end .

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mean = fraction sum over size end = fraction 90 + 100 + ellipsis +270 over 20 end = pounds 162.

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mean = overline x = fraction sum x over n end = fraction 3240 over 20 end = pounds 162.

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fraction 4363 p over 15 end simeq 290.9 p .

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x subscript open bracket 1 close bracket end = 240 and x subscript open bracket 15 close bracket end = 340.

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fraction 4306 p over 15 end simeq 287.1 p .

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fraction pounds 3470 over 20 end = pounds 173.5 simeq pounds 174

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Figure 8

A stemplot with 11 levels. The numbers in the stem start at 0, increase in steps of one and end at 10. At level 0 there is one leaf, 7. Level 1 has one leaf, 5. Level 2 has no leaves, while level 3 has two leaves, 3, 5. Level 4 has three leaves, 2, 2, 3 and level 5 has two leaves, 5, 8. Level 6 has three leaves, 4, 6, 8. Level 7 has five leaves, 1, 1, 6, 8, 9. Level 8 has nine leaves, 0, 1, 1, 3, 4, 5, 5, 6, 9. Level 9 has five leaves, 1, 1, 3, 5, 9. Level 10 has two leaves which are both 0. Beneath the stemplot is written n = 33, followed by 0 vertical line 7 represents a score of 7 per cent. The vertical lines sits horizontally between the 0 and the 7.

Back to - Figure 8


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fraction 1 over 2 end open bracket pounds 269 + pounds 270 close bracket = pounds 269.5 simeq pounds 270.

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fraction 2326 over 33 end simeq 70.5.

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fraction 7856 over 26 end = 302.1538 simeq 302.2.

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fraction 2480 over 17 end =145.8824 simeq 146.

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Figure 9 Means of biscuit prices

The figure shows a horizontal arrow pointing to the right and labelled pence. Part way in from the left is a marker above which is written 74.0. Farther to the right, but before the end of the arrow, is another marker, above which is written 81.6.

Back to - Figure 9 Means of biscuit prices


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fraction sum open bracket of the combined batch prices close bracket over size open bracket of the combined batch close bracket end .

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mean = fraction sum over size end

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sum = mean times size .

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mean = fraction combined sum over combined size end = fraction 408 +592 over 13 end = fraction 1000 over 13 end simeq 76.9

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overline x subscript uppercase C end = fraction overline x subscript uppercase A end n subscript uppercase A end + overline x subscript uppercase B end n subscript uppercase B end over n subscript uppercase A end + n subscript uppercase B end end comma

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overline x subscript uppercase A end = mean of batch uppercase A comma n subscript uppercase A end = size of batch uppercase A comma overline x subscript uppercase B end = mean of batch uppercase B comma n subscript uppercase B end = size of batch uppercase B.

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overline x subscript uppercase A end = 81.6 comma n subscript uppercase A end = 5 comma overline x subscript uppercase B end = 74.0 comma n subscript uppercase B end = 8.

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overline x subscript uppercase C end = fraction open bracket 81 .6 times 5 close bracket + open bracket 74 .0 times 8 close bracket over 5 +8 end .

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Figure 10 Point of balance at the weighted mean

Point of balance at the weighted mean. The figure shows a horizontal arrow pointing to the right and labelled pence. Three appropriately spaced points are marked on the line: 74.0, 76.9 and 81.6. The distance between the 74.0 and 76.9 is less than the distance between 76.9 and 81.6. Immediately below 76.9 there is a solid arrow head pointing upwards, which is the fulcrum or balancing point of the balance. Suspended from the point marked 74.0 is a pile of 8 discs and suspended from the point marked 81.6 is a pile of 5 discs.

Back to - Figure 10 Point of balance at the weighted mean


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Batch uppercase A has mean 119 and size 7. Batch uppercase B has mean 185 and size 13 .

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overline x subscript uppercase A end = 119 comma n subscript uppercase A end = 7 comma overline x subscript uppercase B end = 185 comma n subscript uppercase B end = 13.

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overline x subscript uppercase C end = fraction open bracket 119 times 7 close bracket + open bracket 185 times 13 close bracket over 7 +13 end = fraction 833 +2405 over 20 end = fraction 3238 over 20 end = 161.9 simeq 162.

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cost = price times quantity .

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fraction open bracket 136 .9 times 41 .2 close bracket + open bracket 148 .0 times 10 close bracket over 41 .2 + 10 end .

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overline p = fraction p sub 1 q sub 1 + p sub 2 q sub 2 over q sub 1 + q sub 2 end .

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. begin 2 by 2 array q sub 1 =10 next column quantity next row p sub 1 = 40 next column price end array } first occasion

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. begin 2 by 2 array q sub 2 =6 next column quantity next row p sub 2 =45 next column price end array } second occasion.

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overline p = fraction open bracket 40 times 10 close bracket + open bracket 45 times 6 close bracket over 10 +6 end = fraction 400 + 270 over 16 end = fraction 670 over 16 end =41.875 simeq 41.9.

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fraction overline x subscript uppercase A end n subscript uppercase A end + overline x subscript uppercase B end n subscript uppercase B end over n subscript uppercase A end + n subscript uppercase B end end and fraction p sub 1 q sub 1 + p sub 2 q sub 2 over q sub 1 + q sub 2 end comma

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fraction x sub 1 w sub 1 + x sub 2 w sub 2 over w sub 1 + w sub 2 end .

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fraction 1 over 2 end open bracket 3.818 + 3.740 close bracket = 3.779.

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fraction 3 .818 q sub 1 + 3 .740 q sub 2 over q sub 1 + q sub 2 end comma

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fraction open bracket 3 .818 times 83 close bracket + open bracket 3 .740 times 4 close bracket over 83 + 4 end .

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fraction sum of { number times weight } over sum of weights end = fraction sum of products over sum of weights end .

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Batch 1 with mean 525.5 and batch size 6. Batch 2 with mean 468.0 and batch size 2. Batch 3 with mean 504.2 and batch size 12.

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fraction sum of products over sum of weights end = fraction 10139 .4 over 20 end = 506.97.

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1.3 with weight 2 1.9 with weight 1 1.7 with weight 3.

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fraction open bracket 1 .3 times 2 close bracket + open bracket 1 .9 times 1 close bracket + open bracket 1 .7 times 3 close bracket over 2 + 1 + 3 end = fraction 2 .6 + 1 .9 + 5 .1 over 6 end = fraction 9 .6 over 6 end = 1.6.

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Figure 11 Point of balance for three means

Point of balance for three means. The figure shows a horizontal arrow pointing to the right. Four suitably spaced points are marked on it, 1.3, 1.6, 1.7 and 1.9. Immediately below 1.6 there is a solid arrow head pointing upwards, which is the fulcrum or balancing point of the balance. Suspended from the point marked 1.3 is a pile of 2 discs, suspended from the point marked 1.7 is a pile of 3 discs and suspended from the point marked 1.9 is a single disc.

Back to - Figure 11 Point of balance for three means


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fraction sum of products open bracket price times weight close bracket over sum of weights end

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fraction sum x w over sum w end .

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fraction 6973 .436 over 1834 end =3.802310 simeq 3.802.

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fraction sum x w over sum w end = fraction 24854 .49 over 1892 end = 13.136623 simeq 13.14.

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Batch uppercase A has mean price pounds 80.7 and batch size 10. Batch uppercase B has mean price pounds 78.5 and batch size 17.

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fraction open bracket 80 .7 times 10 close bracket + open bracket 78 .5 times 17 close bracket over 10 + 17 end = fraction 2141 .5 over 27 end comma

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fraction open bracket 10 .95 times 8 .5 close bracket + open bracket 12 .70 times 6 close bracket over 8 .5 + 6 end = fraction 169 .275 over 14 .5 end comma

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range = uppercase E subscript uppercase U end minus uppercase E subscript uppercase L end comma

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Figure 12 Median and quartiles

An extended V-shaped diagram showing the median and quartiles. The V-shape has two additional arms, one to the left and one to the right, so the shape now resembles a capital letter M, but with the sides sloping. The numbers represented by x subscript 1, x subscript 2 and so on, ending with x subscript 15 are shown.

The shape starts on the left with x subscript 1, followed, slightly higher and to the right by x subscript 2. Following the same line are x subscript 3 and x subscript 4. At that point the V formation begins, so the next entry x subscript 5 is slightly lower down and to the right and level with x subscript 3. The same line is followed by x subscript 6, x subscript 7, which is level with x subscript 1, and x subscript 8, which is lower than x subscript 1. At x subscript 8 the direction changes again and the letters move slightly to the right and begin to rise, so x subscript 9 is level with x subscript 7. They continue to rise until x subscript 12 is reached. This is level with x subscript 4. The letters then begin to fall again, moving to the right each time, ending with x subscript 15. This is on the same horizontal level as x subscript 1.

There are three clouds, each containing words. The first contains the words ‘Lower quartile’ and has an arrow pointing to the label x subscript 4, which is at the top of the first line. To the right and at the same level is another cloud, containing the words ‘Upper quartile’, from which an arrow points to the label x subscript 12. The third cloud contains the word ‘Median’. From it an arrow points to the lowest label, x subscript 8, in the middle of the diagram.

Back to - Figure 12 Median and quartiles


Figure 13 Quartiles for sample size n=17

Diagram showing the quartiles for sample size n = 17. The V-shape has two additional arms, one to the left and one to the right, so the shape now resembles a capital letter M, but with the sides sloping. The numbers represented by x subscript 1, x subscript 2 and so on, ending with x subscript 17 are shown.

The shape starts on the left with x subscript 1, followed, slightly higher and to the right by x subscript 2, Following the same line are x subscript 3 and x subscript 4. At that point the V formation begins, but there is a gap at the top. The next entry x subscript 5 is level with but to the right of x subscript 4. The same line is followed by x subscript 6, x subscript 7 and x subscript 8, which is level with x subscript 1. Continuing in the same direction leads to x subscript 9, which is at a lower level than x subscript 1. At x subscript 9 the direction changes again and the letters begin to rise, moving slightly to the right again until x subscript 13 is reached. This is level with x subscript 5. X subscript 14 is to the right of and level with x subscript 13 and forms the start of the last slope. The letters then begin to fall again, moving to the right each time, ending with x subscript 17. This is on the same horizontal level as x subscript 1.

There are three clouds, each containing words. The first contains the words ‘Lower quartile’ and has an arrow pointing to the gap between x subscript 4 and x subscript 5. To the right and at the same level is another cloud, containing the words ‘Upper quartile’, from which an arrow points to the gap between x subscript 13 and x subscript 14. The third cloud contains the word ‘Median’. From it an arrow points to the lowest label, x subscript 9, in the middle of the diagram.

Back to - Figure 13 Quartiles for sample size n=17


Figure 14 Quartiles for sample size n=18

Diagram showing the quartiles for sample size n = 18. The V-shape has two additional arms, one to the left and one to the right, so the shape now resembles a capital letter M, but with the sides sloping. The numbers represented by x subscript 1, x subscript 2 and so on, ending with x subscript 18 are shown.

The shape starts on the left with x subscript 1, followed, slightly higher and to the right by x subscript 2, Following the same line are x subscript 3 and x subscript 4. At that point the V formation begins, but there is a gap at the top. The next entry x subscript 5 is level with but to the right of x subscript 4. The same line is followed by x subscript 6, x subscript 7 and x subscript 8, which is level with x subscript 1. Continuing in the same direction leads to x subscript 9, which is at a lower level than x subscript 1. At x subscript 9 there is a gap. The direction changes again and the letters begin to rise. Slightly to the right but level with x subscript 9 is x subscript 10. They continue to rise and move to the right until x subscript 14 is reached. This is level with x subscript 5. There is then a gap and the letters then begin to fall again, starting with x subscript 15 which is level with x subscript 14. They continue down, moving to the right each time, ending with x subscript 18. This is on the same horizontal level as x subscript 1.

There are three clouds, each containing words. The first contains the words ‘Lower quartile’ and has an arrow pointing to the gap between x subscript 4 and x subscript 5 with the arrowhead closer to x subscript 5 than x subscript 4. To the right and at the same level is another cloud, containing the words ‘Upper quartile’, from which an arrow points to the gap between x subscript 14 and x subscript 15 with the arrowhead closer to x subscript 14 than x subscript 15. Below the diagram a third cloud contains the word ‘Median’. From it an arrow points to the gap between x subscript 9 and x subscript 10, in the middle of the diagram.

Back to - Figure 14 Quartiles for sample size n=18


Figure 15 Quartiles for sample size n=20

Diagram showing the quartiles for sample size n = 20. The V-shape has two additional arms, one to the left and one to the right, so the shape now resembles a capital letter M, but with the sides sloping. The numbers represented by x subscript 1, x subscript 2 and so on, ending with x subscript 20 are shown.

The shape starts with x subscript 1, followed, slightly higher and to the right by x subscript 2, Following the same line are x subscript 3, x subscript 4 and x subscript 5. At that point the V formation begins, but there is a gap at the top. The next entry x subscript 6 is level with but to the right of x subscript 5. The same line is followed by x subscript 7, x subscript 8, x subscript 9 and x subscript 10, which is level with x subscript 1. At x subscript 10 there is a gap. The direction changes again and the letters begin to rise. Slightly to the right but level with x subscript 10 is x subscript 11. The letters continue to rise and move to the right until x subscript 15 is reached. This is level with x subscript 6. There is then a gap and the letters move to the right and begin to fall again, starting with x subscript 16 which is level with x subscript 15. They continue down, moving to the right each time, ending with x subscript 20. This is on the same horizontal level as x subscript 11.

There are three clouds, each containing words. Above the diagram the first cloud contains the words ‘Lower quartile’ and has an arrow pointing to the gap between x subscript 5 and x subscript 6 with the arrowhead closer to x subscript 5 than x subscript 6. To the right and at the same level is another cloud, containing the words ‘Upper quartile’, from which an arrow points to the gap between x subscript 15 and x subscript 16 with the arrowhead closer to x subscript 16 than x subscript 15. The third cloud is below the diagram and contains the word ‘Median’. From it an arrow points to the gap between x subscript 10 and x subscript 11, in the middle of the diagram.

Back to - Figure 15 Quartiles for sample size n=20


Figure 16 Prices of flat-screen televisions with a screen size of 24 inches or less

Stemplot with 10 levels, though there are repeated numbers in the stem. The numbers in the stem are 0, 1, 1, 1, 1, 1, 2, 2, 2, 2. At level 0 there is one leaf 9. At the first level 1 there is one leaf, 0. At the second level 1 there are four leaves 2, 3, 3, 3. The third level 1 has five leaves, 4, 5, 5, 5, 5. The fourth level 1 has three leaves, 6, 6, 7. The fifth and last level 1 has three leaves, 8, 8, 9. The first level 2 and the second level 2 have no leaves. The third level 2 has two leaves, 4, 5. The last level 2 has one leaf, 7. Beneath the stemplot is written n = 20, followed by 0 vertical line 9 represents 90 pounds sterling. The vertical line sits horizontally between the 0 and the 9.

Back to - Figure 16 Prices of flat-screen televisions with a screen size of 24 inches or less


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53 60 65 70 70 74 79 81 85 90

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Figure 17 Stemplot of 15 coffee prices

A stemplot with 11 levels, which start at 26 and end at 36. At level 26 there are five leaves 8, 8, 8, 8, 9. At level 27 there two leaves 5, 9. At level 28 there are no leaves. Level 29 has five leaves, 5, 5, 5, 5, 9. Level 30 has one leaf, 5 and level 31 also has one leaf, 5. Levels 32, 33, 34, and 35 have no leaves. Level 36 has one leaf, 9. Beneath the stemplot is written n = 15, followed by 26 vertical line 8 represents 268 pence. The vertical line sits horizontally between the 26 and the 8.

Back to - Figure 17 Stemplot of 15 coffee prices


Figure 18 Stemplot of 14 gas prices

A stemplot with 8 levels, which start at 374 and end at 381. Level 374 has three leaves, 0, 0, 3. Level 375 has no leaves. Level 376 has two leaves, 0, 7. Level 377 has one leaf, 6. Level 378 also has one leaf, 4. Level 379 has two leaves, 5, 6. Level 380 has four leaves, 1, 1, 4, 5. Level 381 has one leaf, 8. Beneath the stemplot is written n = 14, followed by 374 vertical line 0 represents 3.740 pence per kilowatt hour. The vertical line sits horizontally between the 374 and the 0.

Back to - Figure 18 Stemplot of 14 gas prices


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uppercase Q sub 1 = 3.743 + fraction 3 over 4 end open bracket 3.760 minus 3.743 close bracket = 3.75575 simeq 3.756

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uppercase Q sub 3 = 3.801 + fraction 1 over 4 end open bracket 3.804 minus 3.801 close bracket = 3.80175 simeq 3.802.

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IQR = uppercase Q sub 3 minus uppercase Q sub 1 .

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IQR = uppercase Q sub 3 minus uppercase Q sub 1 = 180 minus 130 = 50.

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range = uppercase E subscript uppercase U end minus uppercase E subscript uppercase L end = 369 p minus 268 p = 101 p .

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IQR = uppercase Q sub 3 minus uppercase Q sub 1 = 299 p minus 268 p = 31 p .

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IQR = uppercase Q sub 3 minus uppercase Q sub 1 =3.80175 minus 3.75575 =0.046 comma

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uppercase E subscript uppercase U end minus uppercase E subscript uppercase L end = 325 p minus 268 p = 57 p comma

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Figure 19 Values in a five-figure summary

Values in a five-figure summary. There are four lines, forming the shape of a letter M with sloping sides. At the base of the line on the left is capital E subscript capital L, indicating the lower extreme, the lowest value in the data. At the top of that line is capital Q subscript 1, indicating the lower quartile. The line then slopes down and to the right. At the end of the second line is capital M, indicating the median. This is on the same horizontal level as capital E subscript capital L. The line then rises and slopes to the right. It ends at capital Q subscript 3, indicating the upper quartile, which is at the same horizontal level as capital Q subscript 1. The line then falls, sloping to the right. At the end of the line is capital E subscript capital U, indicating the upper extreme, the highest value in the data. This is at the same horizontal level as capital M.

Back to - Figure 19 Values in a five-figure summary


Figure 20

A five-figure summary which is a diagrammatic representation showing the batch size, n, the median capital M, the lower quartile capital Q subscript 1, the upper quartile capital Q subscript 3, the lower extreme capital E subscript capital L, and the upper extreme capital E subscript capital U. The diagram forms three sides of a rectangle, with the bottom line missing. It therefore has a vertical line to the left, a horizontal line across the top and a vertical line to the right. To the left of the left vertical line is written n. Towards the bottom of the line and to its right is written capital E subscript capital L and capital Q subscript 1, with capital Q subscript 1 being above capital E subscript capital L. Beneath the middle of the horizontal line is written capital M. To the left of the second vertical line but level with capital Q subscript 1 is written capital Q subscript 3. Below that, and level with capital E subscript capital L, is written capital E subscript capital U.

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Figure 21

A five-figure summary. The diagram forms three sides of a rectangle, with the bottom line missing. It therefore has a vertical line to the left, a horizontal line across the top and a vertical line to the right. To the left of the left vertical line is written n = 20. Towards the bottom of the left vertical line and to its right is written 90 and above that 130. Beneath the middle of the horizontal line is written 150. To the left of the second vertical line and level with 130 is written 180. Below that and level with 90 is written 270.

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Figure 22 Boxplot of batch of 14 gas prices

A boxplot with a horizontal scale with an arrow head, pointing right, at the right-hand end. It is labelled pence per kilowatt hour. The scale starts just before the first marked point of 3.74 and is then marked in four intervals of 0.02, ending with 3.82. The boxplot is drawn above and parallel to the line. The first whisker starts at the lower extreme, level with 3.74.The box starts at the lower quartile, 3.756 and ends at the upper quartile, 3.802. Within the box but nearer the right-hand end is a vertical line indicating the median at 3.790. The second whisker starts at the midpoint of the right-hand end of the box and stretches to 3.818, as far as the upper extreme.

Back to - Figure 22 Boxplot of batch of 14 gas prices


Figure 23 A standard boxplot with annotation

A boxplot which consists of a line, a rectangle, known as a box, and a second line. The first line, known as a whisker, starts at capital E subscript capital L, the lower extreme and leads to the midpoint of the left side of the box. Immediately above and at the corner of the box is written capital Q subscript 1. This point is the lower quartile. The box stretches for some way to the right, ending at the upper quartile, with capital Q subscript 3 written above the end of the box and on the same horizontal level as capital Q subscript 1. A second horizontal line, also known as a whisker, starts at the midpoint of the right-hand end of the box and stretches as far as capital E subscript capital U, the upper extreme. The median is marked by capital M above the box at the appropriate position and by a vertical line within the box. Beneath the boxplot are 4 horizontal curly brackets, each of which has 25% written under it. The first stretches from immediately beneath capital E subscript capital L to the start of the box. The second bracket stretches from the start of the box to level with the vertical line level indicating the position of the median. The third stretches from the line level with the median to level with capital Q subscript 3 and the last stretches from level with capital Q subscript 3 to level with capital E subscript capital U.

Back to - Figure 23 A standard boxplot with annotation


Figure 24 Boxplot of batch of 20 television prices

The horizontal scale is marked from just before 100 to 275 in intervals of 25 units. The scale is labelled pounds sterling. The boxplot shows that the lower extreme is less than 100. The whisker leads to the lower quartile, at the start of the box. This occurs just past 125. The box contains a vertical line, indicating the position of the median. This occurs at 150. The box ends at the upper quartile, which occurs just after 175. The right-hand whisker ends at 250. There is then a gap. Farther on, but at the same level as the whisker, is an asterisk.

Back to - Figure 24 Boxplot of batch of 20 television prices


Figure 25 Stemplot of 14 gas prices

A stemplot with 8 levels, which start at 374 and end at 381. Level 374 has three leaves, 0, 0, 3. Level 375 has no leaves. Level 376 has two leaves, 0, 7. Level 377 has one leaf, 6. Level 378 also has one leaf, 4. Level 379 has two leaves, 5, 6. Level 380 has four leaves, 1, 1, 4, 5. Level 381 has one leaf, 8. Beneath the stemplot is written n = 14, followed by 374 vertical line 0 represents 3.740 pence per kilowatt hour. The vertical line lies horizontally between the 374 and the 0.

Back to - Figure 25 Stemplot of 14 gas prices


Figure 26

A five-figure summary. The diagram forms three sides of a rectangle, with the bottom line missing. It therefore has a vertical line to the left, a horizontal line across the top and a vertical line to the right. To the left of the left vertical line is written n = 14. Towards the bottom of the line and to the right is written 3.740 and above that 3.756. Beneath the middle of the horizontal line is written 3.790. To the left of the second vertical line and level with 3.756 is written 3.802. Below that, level with 3.740, is written 3.818.

Back to - Figure 26


Figure 27 Boxplot of batch of 14 gas prices

Boxplot of batch of 14 gas prices, previously shown as Figure 17. There is a horizontal scale with an arrow head, pointing right, at the right-hand end. It is labelled pence per kilowatt hour. The scale starts just before the first marked point of 3.74 and is then marked in four intervals of 0.02, ending with 3.82. The boxplot is drawn above and parallel to the line. The first whisker starts at the lower extreme, level with 3.74.The box starts at the lower quartile, 3.756 and ends at the upper quartile, 3.802. Within the box but nearer the right-hand end is a vertical line indicating the median at 3.790. The second whisker starts at the midpoint of the right-hand end of the box and stretches to 3.818, as far as the upper extreme.

Back to - Figure 27 Boxplot of batch of 14 gas prices


Figure 28 Stemplot of ten camera prices

A stemplot with 9 levels, which are 5, 5, 6, 6, 7, 7, 8, 8, 9. The first level 5 has one leaf, 3. The second level 5 has no leaves. The first level 6 has one leaf, 0. The second level 6 has one leaf, 5. The first level 7 has three leaves, 0, 0, 4. The second level 7 has one leaf, 9. The first level 8 has one leaf, 1 and the second level 8 also has one leaf, 5. Level 9 has one leaf, 0. Beneath the stemplot is written n = 10, followed by 5 vertical line 3 represents 53 pounds sterling. The vertical line sits horizontally between the 5 and the 3.

Back to - Figure 28 Stemplot of ten camera prices


Figure 29 Boxplot of batch of ten camera prices

A boxplot with a horizontal scale with an arrow head, pointing right, at the right-hand end. It is labelled pounds sterling. The scale starts just before the first marked point of 50 and then marked in five intervals of 10, ending with 90. The boxplot is drawn above and parallel to the line. The first whisker starts with capital E subscript capital L at 53 and ends at capital Q subscript 1, the lower quartile, 65. The box ends at capital Q subscript 3, the upper quartile, 81. Near the centre of the box is a vertical line indicating the median at 72. The second whisker starts at the midpoint of the right-hand end of the box and stretches to capital E subscript capital U at 90.

Back to - Figure 29 Boxplot of batch of ten camera prices


Figure 30 Stemplot of arithmetic stores

A stemplot with 11 levels. The numbers in the stem start at 0, increase in steps of one and end at 10. At level 0 there is one leaf, 7. Level 1 has one leaf, 5. Level 2 has no leaves, while level 3 has two leaves, 3, 5. Level 4 has three leaves, 2, 2, 3. Level 5 has two leaves, 5, 8. Level 6 has three leaves, 4, 6, 8. Level 7 has five leaves, 1, 1, 6, 8, 9. Level 8 has nine leaves, 0, 1, 1, 3, 4, 5, 5, 6, 9. Level 9 has five leaves, 1, 1, 3, 5, 9. Level 10 has two leaves which are both 0. Beneath the stemplot is written n = 33, followed by 0 vertical line 7 represents a score of 7 per cent.

Back to - Figure 30 Stemplot of arithmetic stores


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uppercase Q sub 1 = fraction 1 over 2 end open bracket 55+58 close bracket % = 56.5 % simeq 57 % .

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uppercase Q sub 3 = fraction 1 over 2 end open bracket 86+89 close bracket % = 87.5 % simeq 88 % .

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uppercase Q sub 3 minus uppercase Q sub 1 = 87.5 % minus 56.5 % = 31 % .

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uppercase Q sub 1 = pounds 229 + fraction 3 over 4 end open bracket pounds 230 minus pounds 229 close bracket = pounds 229.75 simeq pounds 230.

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uppercase Q sub 3 = pounds 320 + fraction 1 over 4 end open bracket pounds 349 minus pounds 320 close bracket = pounds 327.25 simeq pounds 327.

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uppercase Q sub 3 minus uppercase Q sub 1 = pounds 327.25 minus pounds 229.75 = pounds 97.5 simeq pounds 98.

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Figure 31 Five-figure summary of arithmetic scores

A five-figure summary. The diagram forms three sides of a rectangle, with the bottom line missing. It therefore has a vertical line to the left, a horizontal line across the top and a vertical line to the right. To the left of the left vertical line is written n = 33. Towards the bottom of the line and to its right is written 7 and above that 57. Beneath the middle of the horizontal line is written 79. To the left of the second vertical line and level with 57 is written 88. Below that and level with 7 is written 100.

Back to - Figure 31 Five-figure summary of arithmetic scores


Figure 32 Five-figure summary of television prices

A five-figure summary. The diagram forms three sides of a rectangle, with the bottom line missing. It therefore has a vertical line to the left, a horizontal line across the top and a vertical line to the right. To the left of the left vertical line is written n = 26. Towards the bottom of the line and to its right is written 170 and above that 230. Beneath the middle of the horizontal line is written 270. To the left of the second vertical line and level with 230 is written 327. Below that and level with 170 is written 699.

Back to - Figure 32 Five-figure summary of television prices


Figure 33 Boxplot of batch of 33 arithmetic scores

A boxplot of a batch of 33 arithmetic scores. There is a horizontal scale with an arrow head, pointing right, at the right-hand end. It is labelled percentage. The scale starts at 0 and is then marked in five intervals of 20, ending with 100. Above the line, and level with the boxplot is an asterisk at 7. The first whisker on the boxplot starts at 15. The capital Q subscript 1 is at about 58, capital M is at about 79 and capital Q subscript 3 is at about 86. The second whisker stretches to the upper extreme at 100.

Back to - Figure 33 Boxplot of batch of 33 arithmetic scores


Figure 34 Boxplot of batch of 26 television prices

A boxplot of a batch of 26 television prices. There is a horizontal scale with an arrow head, pointing right, at the right-hand end. It is labelled pounds sterling. The scale starts at 100 and is then marked in six intervals of 100, ending with 700. The first whisker on the boxplot starts at 170. The box starts at capital Q subscript 1, which is about 230 and capital M is at about 270. The box ends at capital Q subscript 3 at about 320. The second whisker ends at 429. There are two asterisks. One is at 649 and the other is at 699.

Back to - Figure 34 Boxplot of batch of 26 television prices


Figure 35 A chained index

A chained index. There are two rows. The first is labelled Index value. There is then the number 100, followed by 7 sets of 3 dots. Above the 100 is an arrow, leading to the first set of dots. Then from that set is another arrow linking it to the next set, and so on to the end of the line. This gives 7 arrows in total. The next row is labelled Year. There is then a chain of 7 oval shapes, each overlapping slightly with the next one. The oval under the 100 in the first row contains the year 2007, the next one, under the first set of dots, contains 2008. The pattern continues in this way, the year increasing by 1 each time. The last entry, under the sixth set of dots, contains 2013, though it is clear that the numbers can continue further. Under the table is a cloud containing the words ‘2007 is the base year’. From it an arrow points to 2007 in the first link of the chain.

Back to - Figure 35 A chained index


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open bracket value of the index in 2007 comma which is taken as 100 close bracket times fraction gas price in 2008 over gas price in 2007 end .

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open bracket value of the index in 2007 comma which is 100 close bracket times open bracket electricity price ratio for 2008 relative to 2007 close bracket = 100 times 1.145 = 114.5.

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quantity used = fraction expenditure over price end .

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fraction 9298 over 24 end simeq 387.4.

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pounds open bracket 11234.6 + 3671.4 close bracket = pounds 14906.0.

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fraction 14906 .0 over 12503 end simeq 1.192.

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2008 index = 100 times 1.192 = 119.2.

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fraction open bracket 1 .208 times 9298 close bracket + open bracket 1 .145 times 3205 close bracket over 9298 +3205 end = fraction 14901 .709 over 12503 end simeq 1.192 comma

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fraction 30 over 29 end simeq 1.034.

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fraction 98 over 87 end simeq 1.126.

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fraction open bracket 1 .034 times 8145 close bracket + open bracket 1 .126 times 2991 close bracket over 8145 +2991 end = fraction 11789 .796 over 11136 end simeq 1.059.

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fraction open bracket 1 .034 times 23733 close bracket + open bracket 1 .126 times 2275 close bracket over 23733 +2275 end = fraction 27101 .572 over 26008 end simeq 1.042.

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119.2 times 1.059 simeq 126.2.

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Figure 36 Determining a chained price index

Determining a chained price index. There are 3 rows. The first is labelled Price ratios, the second is labelled Index and the third is labelled Base year. The row headed Index contains three numbers. These are 100, 119.2 and 126.2. Above these, the row labelled Price ratios contains two arrows. The first goes from the index number 100 to the index number 119.2. Above the arrow is written times 1.192. The second arrow goes from the index number 119.2 to the index number 126.2. Above the arrow is written times 1.059. The last row has three ovals each containing a number. The first contains 2007, below the index number 100. The second contains 2008, below the index number 119.2. The third and last contains 2009 below the index number 126.2.

Back to - Figure 36 Determining a chained price index


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fraction price that year over price previous year end .

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value of index for previous year times all minus commodities price ratio .

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fraction 28 over 30 end simeq 0.933.

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fraction 88 over 98 end simeq 0.898.

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fraction open bracket 0 .933 times 23733 close bracket + open bracket 0 .898 times 2275 close bracket over 23733 +2275 end = fraction 24185 .839 over 26008 end simeq 0.930.

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126.2 times 0.930 simeq 117.4.

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fraction 30 over 28 end simeq 1.071.

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fraction 86 over 88 end simeq 0.977.

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fraction open bracket 1 .071 times 23969 close bracket + open bracket 0 .977 times 2920 close bracket over 23969 +2920 end = fraction 28523 .639 over 26889 end simeq 1.061.

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117.4 times 1.061 simeq 124.6.

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RPI for Nov. 2011 = RPI for Jan. 2011 times open bracket price ratio for Nov. 2011 relative to Jan. 2011 close bracket .

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CPI for Nov. 2011 = CPI for Dec. 2010 times open bracket price ratio for Nov. 2011 relative to Dec. 2010 close bracket .

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Figure 37 Structure of the RPI in 2012 (based on data from the Office for National Statistics)

Two concentric circles showing the structure of the Retail Prices Index in 2012, based on data from www.ons.gov.uk. The inner circle shows 5 sectors, representing the 5 fundamental groups of goods and services, each with their own colour. The biggest sector, making an angle of about 150 degrees at the centre, is coloured green and represents Housing and household expenditure. Reading round the circle clockwise from that sector there is a small sector, making an angle of about 30 degrees, coloured red and labelled Personal expenditure. The next sector, making an angle of approximately 90 degrees at the centre, is coloured yellow and labelled Travel and leisure. The next sector, making an angle of approximately 60 degrees at the centre, and coloured dark blue, is labelled Food and catering. The last sector, making an angle of about 30 degrees at the centre is labelled Alcohol and tobacco. The outer circle forms a concentric ring round the inner circle. Each of the sectors in the inner circle is subdivided in the ring, to show a breakdown of the relevant type of expenditure. The colour coding continues, but the colours in the outer ring are paler than those in the inner circle.

Housing and household expenditure is divided into four. These are 1 housing, 2 fuel and light, 3 household goods and 4 household services, with housing accounting for about half the expenditure in this sector. Personal expenditure is split into two approximately equal parts, 1 clothing and footwear and 2 personal goods and services. Travel and leisure is split into four parts. The first and largest of these, accounting for about half the expenditure in this sector, is labelled motoring expenditure. The second is fares and other travel costs and the third is leisure goods, which together amount to about the same expenditure as the fourth part, leisure services. ‘Food and catering’ is divided into two, separating food from catering, with the part labelled catering being less than half that of food. Alcohol and tobacco is split between the two, with alcohol being almost twice the size of tobacco.

Back to - Figure 37 Structure of the RPI in 2012 (based on data from the Office for National Statistics)


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fraction 1 over 10 end times pounds 540 = pounds 54 .

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fraction 2 over 5 end times pounds 540 = pounds 216 simeq pounds 220 .

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fraction 1 over 4 end times pounds 540 = pounds 135 simeq pounds 140 .

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Figure 38 A checklist for one household’s average monthly expenditure

A checklist for one household’s average monthly expenditure. An opportunity to compare your monthly expenditure with that of a two-person household. The table consists of the RPI groups and subgroups in the first column. There are then 3 columns under a general heading ‘Expenditure and weights’. The columns are headed ‘Expenditure 2012 in pounds sterling’, ‘Group totals in pounds sterling’ and ‘Group weights’. There is then a vertical line and there are three more columns under the general heading ‘Your expenditure and weights’. These 3 columns are also headed ‘Expenditure 2012 in pounds sterling’, ‘Group totals in pounds sterling’ and ‘Group weights’ and there are lines to indicate where entries should be made. However, there are no entries in these three columns. They have been left blank for you to input your own data.

The entries for the two-person data are as follows. Under the group heading Food and catering there are 3 subgroups. At home has an expenditure of 370. Canteen, snacks and takeaways has an expenditure of 80 and restaurant meals has an expenditure of 20. Together these give a group total of 470, which is written in the next column, on the row below. Level with this and under the column headed Group weights is written 266.

In a similar way, under the group heading Alcohol and tobacco there are 2 subgroups. Alcoholic drink has an expenditure of 8 and cigarettes and tobacco has no expenditure. The group total is therefore 8 and the weight is given as 5.

Under the group heading Housing and household expenditure there are 10 subgroups. Mortgage interest forward slash rent has an expenditure of 82. Council tax has an expenditure of 95. Water charges have an expenditure of 47 and household insurance has an expenditure of 29. Repairs, maintenance or DIY has an expenditure of 40. Gas, electricity, coal or oil bills has an expenditure of 210.

Household goods, which includes furniture, appliances and consumables etc has an expenditure of 70. Telephone and internet cost 20. There was no expenditure for school and university fees or for pet care. In total, this group Housing and household expenditure had a group total of 593 and a group weight of 336.

Under the group heading Personal expenditure clothing and footwear had an expenditure of 45 and other, which included hairdressing, chemists’ goods etc, had expenditure of 10. Together they have a group total of 55 and a group weight of 31.

There are 8 subgroups under the group Travel and leisure. Motoring, which includes purchase, maintenance, petrol, tax and insurance, had expenditure of 210. Fares had expenditure of 200. Books, newspapers and magazines had expenditure of 80. Audio-visual equipment, CDs etc had expenditure of 15. Toys, photographic and sports goods had expenditure of 3, while TV purchase or rental and licence had no expenditure. Cinema, theatre etc had expenditure of 30 and holidays had expenditure of 100. Together these give a group total of 638 and a weight of 362.

Under the group totals is a total for all the groups, which is 1764 and under the group weights the total is 1000.

Back to - Figure 38 A checklist for one household’s average monthly expenditure


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470 times fraction 1000 over 1764 end simeq 266.

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fraction 470 over 1764 end simeq 0.266.

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0.266 times 1000=266.

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fraction price in November 2011 over price in January 2011 end .

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RPI for month x = open bracket RPI for previous January close bracket times open bracket all minus item price ratio for month x close bracket

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all minus item price ratio = fraction sum of products open bracket price ratio times weight close bracket over sum of weights end = fraction sum r w over sum w end .

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fraction 1041 .624 over 1000 end = 1.041624.

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RPI for Nov. 2011 = RPI for Jan. 2011 times open bracket all minus item price ratio for Nov. 2011 close bracket = 229.0 times 1.041624 = 238.531896 simeq 238.5.

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fraction sum r w over sum w end .

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RPI for month x = RPI for previous January times open bracket all minus item price ratio for month x close bracket .

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sum open bracket w close bracket = comma sum of products open bracket r w close bracket = comma all minus item price ratio = fraction sum of products open bracket r w close bracket over sum open bracket w close bracket end = comma value of RPI in July 2011 = .

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sum open bracket w close bracket =1000 comma sum of products open bracket r w close bracket =1024.608 comma

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all minus item price ratio = fraction sum of products open bracket r w close bracket over sum open bracket w close bracket end = fraction 1024 .608 over 1000 end =1.024608 comma

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value of RPI in July 2011 =229.0 times 1.024608 =234.635232 simeq 234.6.

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fraction value of CPI in February 2012 over value of CPI in February 2011 end = fraction 121 .8 over 117 .8 end simeq 1.034.

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fraction value of RPI in February 2012 over value of RPI in February 2011 end = fraction 239 .9 over 231 .3 end simeq 1.037 comma

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pounds 450 times fraction 239 .9 over 231 .3 end open bracket i.e. pounds 466.73 close bracket per month

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pounds 120 times fraction 121 .2 over 115 .6 end simeq pounds 125.81.

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fraction value of RPI at date B over value of RPI at date A end times 100.

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fraction 231 .3 over 239 .9 end times 100 p = 96.41517 p .

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fraction 100 over 239 .9 end times 100 p .

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fraction 223 .6 over 212 .8 end simeq 1.051 comma

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fraction 212 .8 over 223 .6 end times 100 p simeq 95 p .

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fraction 238 .0 over 225 .8 end simeq 1.054 comma

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fraction 225 .8 over 238 .0 end times 100 p simeq 95 p .

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fraction 232 .5 over 220 .7 end simeq 1.053 comma

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fraction 220 .7 over 232 .5 end times 100 p simeq 95 p .

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sum w =1000 comma sum r w=1007.760 comma

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all minus item price ratio = fraction sum r w over sum w end = fraction 1007 .760 over 1000 end =1.007760 comma

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value of RPI in February 2012 =238.0 times 1.007760 =239.84688 simeq 239.8.

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fraction 225 .8 over 216 .0 end simeq 1.045 comma

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fraction 216 .0 over 225 .8 end times 100 p simeq 96 p .

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fraction 229 .0 over 217 .9 end simeq 1.051 comma

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fraction 217 .9 over 229 .0 end times 100 p simeq 95 p .

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pounds 800 times fraction 234 .4 over 222 .8 end simeq pounds 842 per month .

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