Statistical skills
What the paper asks you to calculate
- Statistical skills are one of the five named skills in the standard, and at Level 2 they stay simple: mean, median, mode, range, plus percentages, percentage change and ratios.
- A calculator is advised equipment. Bring one.
- Every statistic must carry its unit, and most should carry the number of cases it was calculated from.
Mean, median, mode, range
- Mean — add all the values and divide by how many there are. It is the average, and it uses every value.
- Median — put the values in order and take the middle one. With an even number of values, take the mean of the middle two.
- Mode — the value that occurs most often. A set can have no mode, or more than one.
- Range — the highest value minus the lowest. It measures spread, not typical size, and it is decided entirely by the two extreme values.
Which measure to use, and when the mean misleads
- Use the mean when the values are fairly evenly spread and you want a figure that reflects all of them.
- Use the median when a few extreme values would distort the mean. Incomes, house prices and farm sizes almost always need the median.
- Use the mode for categories rather than measurements — the commonest land use, the most frequent wind direction.
- The mean is dragged toward outliers because it uses every value equally. One very large value pulls it up; the median barely moves, because the middle of the order barely moves.
- Quote both when they differ a lot. A mean well above the median is itself evidence of a skewed distribution, and saying so is a geographic observation, not a statistical aside.
Percentages and percentage change
- Percentage of a total: part divided by whole, times 100. Always say what the whole was.
- Percentage change: (new value minus old value) divided by the old value, times 100.
- The old value is the denominator. Dividing by the new value is the commonest error in this calculation.
- A negative answer is a decrease, and should be reported as one.
- Percentage change from a small base exaggerates. Two visitors becoming four is a 100 per cent increase and two people.
- Percentage points are not per cent. Unemployment rising from 4 per cent to 6 per cent is a rise of 2 percentage points, which is a 50 per cent increase in the rate.
Ratios and rank
- A ratio compares two quantities: 3 : 1 means three of the first for every one of the second.
- Simplify by dividing both sides by the smaller number, and round sensibly. About 5 : 1 is usually more useful than 4.87 : 1.
- Rank puts places in order on one indicator. It is useful for comparison and hides the size of the gaps — first and second may be far apart or nearly identical.
Using a statistic as evidence
- A statistic supports a claim; it is not a claim. The mean is 84 mm answers nothing on its own.
- Attach it to the geography: the mean summer rainfall of 84 mm is less than half the winter figure, which is why the river is at its lowest when demand for irrigation is highest.
- Say how many cases the figure comes from. A mean of six values and a mean of six hundred are different kinds of evidence.
- Round sensibly and consistently. Reporting a mean to four decimal places from data measured to the nearest whole number claims a precision the data does not have.
Worked Example
Worked Example
An invented survey records the number of vehicles parked at a trail head on nine summer mornings:
12, 15, 15, 18, 22, 24, 27, 31, 96
The 96 was recorded on the morning of a regional cycling event.
Two years earlier, the mean for the same nine mornings was 14 vehicles.
Calculate the mean, median, mode and range. State which measure best represents a typical morning, and calculate the percentage change in the mean since the earlier survey.
Answer:
Step 1 — the mean.
Add the values:
12 + 15 + 15 + 18 + 22 + 24 + 27 + 31 + 96 = 260
Divide by the number of values, which is 9:
260 divided by 9 = 28.9 vehicles (to one decimal place)
Step 2 — the median.
The values are already in order, and there are nine of them, so the median is the fifth:
12, 15, 15, 18, 22, 24, 27, 31, 96
The median is 22 vehicles.
Step 3 — the mode.
15 appears twice; every other value appears once. The mode is 15 vehicles.
Step 4 — the range.
96 minus 12 = 84 vehicles.
Step 5 — decide which measure represents a typical morning.
The mean is 28.9, but only two of the nine mornings had more than 28 vehicles. The mean has been dragged upward by the single value of 96, recorded during a regional cycling event, which is not a typical morning.
The median of 22 is the better measure, because it is decided by the middle of the order and is almost unaffected by one extreme value.
Note what the range tells you here. A range of 84 in a data set whose middle value is 22 is itself the evidence that an outlier is present.
Step 6 — percentage change in the mean.
Percentage change = (new minus old) divided by old, times 100.
(28.9 - 14) divided by 14 = 14.9 divided by 14 = 1.064
1.064 x 100 = an increase of about 106 per cent
But state the caution. The earlier mean may not have included an event day. Comparing two means when one contains an outlier overstates the change, and the honest comparison would be median against median.
The answers: