1The idea in one paragraph

A measure of central tendency says where the data sit: the mean (add up, divide by how many), the median (the middle value in order) or the mode (the most common value). A measure of dispersion says how spread out they are: the range, the interquartile range, or the standard deviation, which is roughly the typical distance of a value from the mean. The mean and standard deviation use every value, so they are sensitive to outliers; the median and IQR are not. Your GDC produces all of them at once, and the marks are in entering the data correctly, choosing the right output, and saying what the numbers mean. Finally, add a constant to every value and the centre moves but the spread does not; multiply every value and both scale.

2Mean, median and mode

For five test scores 4, 7, 8, 9, 12:

• the mean is x̄ = (4 + 7 + 8 + 9 + 12) ÷ 5 = 40 ÷ 5 = 8;
• the median is the middle value once the data are in order: 8;
• there is no mode, since no value repeats.

The mean has a physical meaning worth holding on to: it is the balance point. Put a weight at each value on a ruler, and the ruler balances on the mean. That is because the deviations from the mean, x − x̄, add up to zero: the values below pull left exactly as hard as the values above pull right. Figure 1 shows it.

For the median of n ordered values, count to position (n + 1) ÷ 2. With n odd that is a single value. With n even, it lands halfway between two values, and the median is their mean: for 3, 5, 6, 8, 9, 11 it is (6 + 8) ÷ 2 = 7.

Which average? The mean uses every value, which is its strength and its weakness. Nine employees of a small firm earn these monthly amounts in euros (invented data): 2100, 2200, 2300, 2300, 2400, 2600, 2800, 3100, 9500. Figure 2 plots them.

The mean is 29 300 ÷ 9 ≈ €3256, higher than eight of the nine salaries. The median is €2400 and the mode €2300. Remove the 9500 and the mean drops to €2475 while the median barely moves, to €2350. So:

• use the mean for data that are roughly symmetric with no outliers; it uses all the information;
• use the median when the data are skewed or have outliers, because it describes a typical value;
• use the mode for the most common value, which is the only average that makes sense for, say, shoe sizes a shop should stock.

In skewed data the mean is dragged towards the tail: positive skew puts the mean above the median.

3The mean of a frequency table

When values repeat, a frequency table saves writing them out. Each value x appears f times, so it contributes f × x to the total.

x̄ = Σfx ÷ n, where n = Σf · from the formula booklet.

Thirty students' books read last month (invented data, the same as in 4.2):

The formula also runs backwards, which is a Paper 1 favourite. The goals scored by a hockey team in its matches (invented) are 0 goals 5 times, 1 goal k times, 2 goals 8 times, 3 goals 4 times and 4 goals 3 times. The mean is 1.8. Find k.

The same idea finds a missing value: if the mean of 6 numbers is 15, their total is 6 × 15 = 90. Mean times count gives the total, and totals can be added and subtracted where means cannot.

4Grouped data: an estimate of the mean, and the modal class

Once data are grouped, the individual values are gone, so the mean can only be estimated. The guide's method is to treat every value in a class as if it sat at the mid-interval value, the midpoint of the class: (lower boundary + upper boundary) ÷ 2. Then use Σfx ÷ n with those midpoints as x. Figure 3 shows the idea for the park-run times from 4.2.

It is an estimate because the runners in each class are not really all at the midpoint; the hope is that those above and below it roughly cancel. On Paper 2 you enter the midpoints as a list and the frequencies as a second list, and the GDC does the sum.

The modal class is the class with the highest frequency, here 25 ≤ t < 30. The guide uses it only for equal class intervals, as here.

5Measures of dispersion

Two data sets can share a mean and still look nothing alike. In Figure 4, sets A and B both have mean 8, but B's values are far more spread out.

• The range is largest − smallest. Quick, but it depends on the two most extreme values only, so one outlier can double it.
• The interquartile range, IQR = Q₃ − Q₁, is the spread of the middle half. It ignores the extremes, so it pairs naturally with the median.
• The standard deviation, σ, measures the typical distance of the values from the mean. It uses every value, so it pairs with the mean.
• The variance is the square of the standard deviation, σ². It is in squared units, which is why the standard deviation is the one you interpret.

The guide says you find σ and σ² with technology, but one calculation by hand shows what the number is. For 4, 7, 8, 9, 12 with mean 8:

So σ = √( Σ(x − μ)² ÷ n ): square each distance from the mean, average the squares, then take the square root to get back to the original units. Squaring stops the positive and negative deviations cancelling, and it makes values far from the mean count for more. For the sets in Figure 4, σ ≈ 1.29 for A and σ ≈ 3.96 for B.

On the GDC. Enter the values in a list (and frequencies in a second list if you have a table), then run one-variable statistics: on a TI-84 it is STAT, CALC, 1-Var Stats, with the list and, if needed, the frequency list; on a Casio it is the Statistics menu, CALC, 1-VAR, after setting the list and frequency in SET. The screen gives x̄, Σx, n, the quartiles and the median, and two standard deviations:

• σx divides by n. This is the one to use in AA, because the data set is treated as the population.
• sx divides by n − 1. It is used when a sample is meant to estimate a larger population's spread, and AA questions do not ask for it.

For 4, 7, 8, 9, 12 the GDC shows σx = 2.61 and sx = 2.92. Writing 2.92 loses the accuracy mark. For the books table σx = 1.34, and for the grouped park-run times, with midpoints and frequencies entered, σx ≈ 7.40 minutes, an estimate for the same reason the mean was.

Interpreting σ. Always in context: "the times are typically about 7.4 minutes either side of the mean of 31.4 minutes". A larger standard deviation means less consistency. When comparing two groups, compare means for centre and standard deviations for spread, with both numbers quoted.

6Constant changes to every value

If every value in a data set is changed in the same way, you do not need to recalculate from scratch. Think about what happens to the picture, as in Figure 5.

Adding or subtracting a constant b slides every value the same distance. The whole data set moves, so every measure of centre moves by b. The gaps between values do not change, so every measure of spread stays the same.

Multiplying by a constant a (a > 0) stretches the data away from zero. Every value, every measure of centre, and every gap is multiplied by a, so the spread is multiplied by a too. The variance, a squared quantity, is multiplied by a².

Adding moves the centre only. Multiplying scales both centre and spread; the variance scales by the square.

The guide's own two cases: subtract 3 from every value and the mean falls by 3 while the standard deviation is unchanged; double every value and both the mean and the standard deviation double.

Worked example. A week of noon temperatures in a town has mean 15 °C and standard deviation 4 °C. Convert to Fahrenheit with F = 1.8C + 32.

A multiplier can be negative, for example if every value is subtracted from 100 (y = 100 − x). The mean becomes 100 − x̄, and the standard deviation is multiplied by |−1| = 1, because a spread cannot be negative.

7Quartiles of discrete data, and why methods disagree

For a short list, most GDCs find quartiles like this: put the data in order, find the median, split the data into a lower half and an upper half leaving the median out when n is odd, and take the median of each half. With n even there is no middle value to leave out, and the halves are simply the lower and upper n ÷ 2 values.

The guide asks you to know that different methods exist. Another common one keeps the median in both halves when n is odd, and spreadsheet functions use yet another rule that interpolates between values. Figure 6 applies the first two methods to nine values.

Neither is wrong; they are different conventions, and the difference shrinks as the data set grows. In an exam, the GDC's answer is accepted, and a hand answer by a sensible method usually is too. Use one method consistently within a question.