For a batch size of 20, the median position is . So, the median will be halfway between
and
. These are both 150, so the median is £150.
A stemplot of all 14 prices in the table is shown below.
Figure 6 Stemplot of 14 gas prices
Stemplots for the prices for northern and southern cities are shown below.
Figure 7 Stemplots for northern and southern cities separately.
For a batch size of 14, the median position is . So, the all-cities median will be halfway between
and
. These are 3.784 and 3.795, so the median is 3.7895, which is 3.790 when rounded to three decimal places. (The rounded median
should be written as 3.790 and not 3.79, to show it is accurate to three decimal places and not just two.)
For the northern and southern batches, both of size 7, the median for each is the value of (that is,
). This is 3.776 for the northern batch and 3.795 for the southern batch.
The range is the difference between the upper extreme, , and the lower extreme,
(range
). So the all-cities range is
the range for the northern batch is
and the range for the southern batch is
The medians and ranges are summarised below.
Batch | Median | Range |
---|---|---|
All cities |
3.790 |
0.078 |
Northern cities |
3.776 |
0.064 |
Southern cities |
3.795 |
0.075 |
Thus the general level of gas prices in the country as a whole was about 3.790p per kWh. The average price differed by only 0.078p per kWh across the 14 cities.
The difference between the median prices for the northern and southern cities is 0.019p per kWh , with the south having the higher median.
The analysis does not clearly reveal whether the general level of gas prices for typical consumers in 2010 was higher in the south or in the north, though there is an indication that prices were a little higher in the south. The range of prices was also rather greater in the south. It is worth noting that the differences in gas prices between the cities in Table 3 were generally small, when measured in pence per kWh – although, with a typical annual gas usage of 18 000 kWh, the price difference between the most expensive city and the cheapest would amount to an annual difference in bills of about £14 on a typical bill of somewhere around £700.
Using the data for the prices from Activity 1:
Or using the notation,
and
, so
The prices were rounded to the nearest £10, so it is appropriate to keep one more significant figure for the mean, that is, to show it accurate to the nearest £1. So since the exact value is £162, it needs no further rounding.
The completed table is:
Batch | Mean | Median |
---|---|---|
Seven southern cities |
3.7859 |
3.795 |
Five southern cities (excluding Cardiff and Ipswich) |
3.7996 |
3.796 |
Whereas deletion of Cardiff and Ipswich has the effect of increasing the mean price by 0.0137p per kWh, the median price increases by only 0.001p per kWh. This is what we would expect as, in general, the more resistant a measure is, the less it changes when a few extreme values are deleted.
The completed table is:
Batch | Mean | Median |
---|---|---|
Five cities (correct data) |
3.7996 |
3.796 |
Five cities (with misprint) |
4.6996 |
3.796 |
Here the median is completely unaffected by the misprint, although the mean changes considerably.
For the arithmetic scores, the position of the median is , so the median is 79%.
For the television prices, the position of the median is , so the median is halfway between
and
. Thus, the median is
For the batch of arithmetic scores in part (a) of Exercise 1, the sum of the 33 values is 2326 and
Therefore, the mean is 70.5%. (The original data are given to the nearest whole number, so the mean is rounded to one decimal place.)
For the batch of television prices in part (b) of Exercise 1, the sum of the 26 values is 7856 and
Therefore, the mean is £302.2.
For the median, there are now 17 prices left in the batch, so the median is at position . It is therefore 150.
The sum of the remaining 17 values is 2480, so the mean is
In this case, removing the three highest prices has not changed the median at all, but it has reduced the mean considerably. This illustrates that the median is a more resistant measure than the mean.
Back to - Exercise 3 The effect of removing values on the median and mean
You should expect the weighted mean price to be nearer the London price, because of Rule 2 for weighted means (Subsection 2.1) and given that London has a much larger weight then Edinburgh.
The weighted mean price given by the formula in Example 11 is (after rounding) 3.814p per kWh, which is indeed much closer to the London price than to the Edinburgh price.
This is the same as a simple (unweighted) mean of the two scores, because the two component scores have equal weight. It lies
exactly halfway between the two scores ().
This is slightly less than the simple mean in (a) because the component with the lower score (TMA) has the greater weight.
This is slightly higher than the simple mean in (a) because the component with the higher score (iCMA) has the greater weight.
(Note that the weights need not necessarily sum to 100, even when dealing with percentages.)
This is even lower than (b), so even nearer the lower score (TMA), because the TMA score has even greater weight.
This is the same as (d) because the ratios of the weights are the same; they are both in the ratio 1 to 3. That is, (
).
(We say this as follows: ‘the ratio 25 to 75 equals the ratio 30 to 90’.)
The table showing the required sums (and the values in the column, that you may not have had to write down), is as follows.
City | Price (p/kWh): ![]() |
Weight: ![]() |
Price ![]() ![]() |
---|---|---|---|
Aberdeen |
13.76 |
19 |
261.44 |
Belfast |
15.03 |
58 |
871.74 |
Edinburgh |
13.86 |
42 |
582.12 |
Leeds |
12.70 |
150 |
1 905.00 |
Liverpool |
13.89 |
82 |
1 138.98 |
Manchester |
12.65 |
224 |
2 833.60 |
Newcastle-upon-Tyne |
12.97 |
88 |
1 141.36 |
Nottingham |
12.64 |
67 |
846.88 |
Birmingham |
12.89 |
228 |
2 938.92 |
Canterbury |
12.92 |
5 |
64.60 |
Cardiff |
13.83 |
33 |
456.39 |
Ipswich |
12.84 |
14 |
179.76 |
London |
13.17 |
828 |
10 904.76 |
Plymouth |
13.61 |
24 |
326.64 |
Southampton |
13.41 |
30 |
402.30 |
Sum |
1892 |
24 854.49 |
Thus ,
and
So the weighted mean of electricity prices is 13.14p per kWh.
Mean price of all the cameras is
which is £79.3 (rounded to the same accuracy as the original means).
Mean price of all the material is
which is £11.67 (rounded to the nearest penny).
Here, because , an appropriate picture of the data would be Figure 12 (Subsection 3.2). To find the lower and upper quartiles,
and
, of this batch, first find
and
. Therefore
and
.
For this batch, so
and
.
and
So the lower quartile is 3.756 p per kWh and the upper quartile is 3.802p per kWh.
The range is the distance between the extremes:
The interquartile range is the distance between the quartiles:
The quartiles, before rounding, are and
. So
and the interquartile range is 0.046p per kWh.
All the necessary figures have already been calculated. You found the median (3.790) in Activity 2 and the quartiles (,
) in Activity 10. The extremes (
,
) and the batch size (
) are clearly shown in the stemplot.
So the five-figure summary is as follows:
Figure 26
Looking at the stemplot, on the whole the lower values are more spread out, indicating that the data are not symmetric and are left-skew.
The central box of the boxplot again shows left skewness, with the left-hand part of the box being clearly longer than the right-hand part. However, this skewness does not show up in the lengths of the whiskers in this batch – they are both the same length.
For the arithmetic scores, so
and
.
The lower quartile is therefore
The upper quartile is
The interquartile range is
For the television prices, so
and
.
The lower quartile is therefore
The upper quartile is
The interquartile range is
Arithmetic scores:
From the stemplot, ,
and
.
Figure 31 Five-figure summary of arithmetic scores
Television prices:
From the data table, ,
and
.
Figure 32 Five-figure summary of television prices
For the boxplot of arithmetic scores, the left part of the box is longer than the right part, and the left whisker is also considerably longer than the right. This batch is left-skew.
For the boxplot of television prices, the right part of the box is rather longer than the left part. The right whisker is also rather longer than the left, and if one also takes into account the fact that two potential outliers have been marked, the top 25% of the data are clearly much more spread out than the bottom 25%. This batch is right-skew.
Back to - Exercise 8 Boxplots and the shape of distributions
The increase (in £/MWh) is . This is
as a proportion of the 2007 price. That is,
of the 2007 price. Or you might have worked this out by finding that the 2008 price is
of the 2007 price, so that again the increase is 20.8% of the 2007 price.
The 2008 electricity price is of the 2007 price, so that the increase is 14.5% of the 2007 price.
The 2008 value of the electricity price index is
The expenditure on a particular fuel in a particular year can be calculated as . Therefore, if the expenditure and price are known, the quantity used can be calculated as
In 2007, Gradgrind’s gas cost £24 per MWh, and they spent £9298 on gas, so the amount of gas they used in MWh was
The other amounts, in MWh, are found in a similar way, and all are shown in the following table.
Energy type | 2007 | 2008 |
---|---|---|
Gas |
387.4 |
280.9 |
Electricity |
42.2 |
34.4 |
The reason that the expenditures went down is simply that Gradgrind used less of each fuel in 2008 than in 2007.
The gas price ratio for 2009 relative to 2008 is
The electricity price ratio for 2009 relative to 2008 is
(Over this year, electricity prices rose a lot more than gas prices.)
Using the 2009 expenditures for weights instead of the 2008 expenditures, the overall energy price ratio for 2009 relative to 2008 is
This price ratio is considerably less than the one found in part (b).
(Note that if full calculator accuracy is retained throughout the calculations, the price ratio is 1.043 to three decimal places.)
The gas price ratio for 2010 relative to 2009 is
The electricity price ratio for 2010 relative to 2009 is
(Both price ratios are less than 1 because, over this year, Gradgrind’s gas and electricity prices both fell.)
The overall energy price ratio for 2010 relative to 2009 is
Then the value of the index for 2010 is found by multiplying the 2009 value of the index by this overall price ratio, giving
Back to - Activity 18 Gradgrind’s energy price index for 2010
The gas price ratio for 2011 relative to 2010 is
The electricity price ratio for 2011 relative to 2010 is
The overall energy price ratio for 2011 relative to 2010 is
Then the value of the index for 2011 is found by multiplying the 2010 value of the index by this overall price ratio, giving
Back to - Exercise 9 Gradgrind’s energy price index for 2011
What you need to remember here is that the size of an area represents the proportion of expenditure on that class of goods or services. (Also, it is admittedly not very easy to estimate these areas ‘by eye’! Your estimates might quite reasonably differ from those given here.)
The sector for ‘Personal expenditure’ looks as if it is approximately a tenth of the whole inner circle – so approximately a tenth of total expenditure is personal expenditure.
‘Housing and household expenditure’ looks as if it is somewhere between a third and a half of the inner circle – perhaps approximately two fifths – so approximately two fifths of expenditure is on housing and household expenditure.
The area for ‘Housing’ takes up about a quarter of the outer ring, so about a quarter of expenditure is on housing.
The amount spent each week on ‘Personal expenditure’ is approximately
The amount spent each week on ‘Housing and household expenditure’ is approximately
The amount spent each week on ‘Housing’ is approximately
Recall, however, that the weights represent average proportions of expenditure, and the spending patterns of the selected household may differ from those of the ‘typical’ household.
Every household will be different, but think about the reasons for any large differences between your weights and those for the RPI.
Group | Price ratio for July 2011 relative to January 2011: ![]() |
2011 weights: ![]() |
Price ratio ![]() ![]() |
---|---|---|---|
Food and catering |
1.024 |
165 |
168.960 |
Alcohol and tobacco |
1.042 |
88 |
91.696 |
Housing and household expenditure |
1.012 |
408 |
412.896 |
Personal expenditure |
1.053 |
82 |
86.346 |
Travel and leisure |
1.030 |
257 |
264.710 |
Sum |
1000 |
1024.608 |
The RPI is calculated using the price ratio and weight of each item. Since the weights of items change very little from one year to the next, the price ratio alone will normally tell you whether a change in price is likely to lead to an increase or a decrease in the value of the RPI. If a price rises, then the price ratio is greater than one, so the RPI is likely to increase as a result. If a price falls, then the price ratio is less than one, so the RPI is likely to decrease. Therefore, since the price of leisure goods fell, this is likely to lead to a decrease in the value of the RPI. For a similar reason, the increase in the price of canteen meals is likely to lead to an increase in the value of the RPI.
Both changes are likely to be small for two reasons. First, the price changes are themselves fairly small. Second, leisure goods and canteen meals form only part of a household’s expenditure: no single group, subgroup or section will have a large effect on the RPI on its own, unless there is a very large change in its price.
The weight of ‘Leisure goods’ was 33 in 2012 (see Table 12). Since ‘Canteen meals’ is only one section in the subgroup ‘Catering’, which had weight 47 in 2012, the weight of ‘Canteen meals’ will be much smaller than 47. (In fact it was 3.) So the weight of ‘Leisure goods’ is much larger than the weight of ‘Canteen meals’.
Since the weight of ‘Leisure goods’ is much larger than the weight of ‘Canteen meals’, and the percentage change in the prices are not too different in size, the change in the price of leisure goods is likely to have a much larger effect on the value of the RPI as a whole.
The ratio of the two RPI values is
or 103.7%. Therefore the annual inflation rate, based on the RPI was 3.7%. (Note that this is slightly higher than the annual inflation rate measured using the CPI.)
Back to - Activity 23 The annual inflation rate in February 2012
The weekly amount in November 2011 should be
For May 2010, the ratio of the value of the RPI to its value one year earlier is
so the annual inflation rate is 5.1%.
The purchasing power of the pound compared to one year previously is
For October 2011, the ratio of the value of the RPI to its value one year earlier is
so the annual inflation rate is 5.4%.
The purchasing power of the pound compared to one year previously is
For March 2011, the ratio of the value of the RPI to its value one year earlier is
so the annual inflation rate is 5.3%.
The purchasing power of the pound compared to one year previously is
(The published index was 239.9. Again, the difference between this and your calculated value is because the ONS statisticians used more accuracy in their intermediate calculations.)
For October 2010, the ratio of the value of the RPI to its value one year earlier is
so the annual inflation rate is 4.5%.
The purchasing power of the pound compared to one year previously is
For January 2011, the ratio of the value of the RPI to its value one year earlier is
so the annual inflation rate is 5.1%.
The purchasing power of the pound compared to one year previously is
The RPI for April 2011 was 234.4 and the RPI for April 2010 was 222.8. So in April 2011, the pension should be