When there are many numbers, like the scores of a class of 30, they are hard to take in at once, so we sum the data up with a single typical value, a measure of central tendency.
There are three: the mean, all values added and divided by how many there are; the median, the middle value when they are put in order; and the mode, the value that appears most often. Each suits a different kind of data.
Two sets of data can share the same typical value while one is bunched near the mean and the other widely spread, so the variance and the standard deviation, which measure spread, are looked at as well.
The pictures show in particular that a single outlier far from the rest moves the mean a long way while the median hardly changes.
Summing up a lot of data in one number
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The shape of 20 data points on a number line, with their mean, median, mode and standard deviation, as the spread changes
1.5
👀 Why we need a typical value
①After a test, the whole class's scores arrive at once.
②Looking at every number one by one hides the overall picture.
③So we describe the data with one representative number, a measure of central tendency.
④The standard deviation at the top of the picture is calculated directly from the 20 points drawn. Increasing the spread leaves the mean near 70 but makes the standard deviation larger.
Three measures of central tendency
Mean
x̄ = x1 + x2 + ... + xnn
All values added together and divided by how many there are; the most widely used measure.
Median
the middle value when the data are put in order of size
With an even number of values it is the mean of the middle two. Extreme values affect it less.
Mode
the value that appears most often
There can be more than one. It also works for data that are not numbers, such as favourite colours.
💡 Working it out for five scores
①Five students scored 60, 70, 70, 80 and 100.
②The mean is (60 + 70 + 70 + 80 + 100) ÷ 5 = 76.
③The median, the third value in order, is 70.
④70 appears twice, so it is the mode.
⑤The mean 76 is larger than the median 70 because the one high score of 100 pulls the mean up.
Outliers and choosing a measure
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How far the mean and the median move when only the fifth student's reading time changes
80
⚠️ When there is an outlier
①A value far away from the others is called an outlier.
②An outlier drags the mean strongly towards it.
③The median is the value in the middle position, so it hardly depends on how large the outlier is.
④For data with a few very large values, such as incomes, the median gives a truer picture than the mean.
⑤In the picture, raising the fifth value all the way to 200 leaves the median at 70.
Spread: deviation, variance and standard deviation
Deviation
deviation = value − mean
It shows how far each value is from the mean, and the deviations always add up to 0.
Variance
V = (dev)21 + (dev)22 + ... + (dev)2nn
The mean of the squared deviations. Deviations are squared because their plain mean is always 0.
Standard deviation
standard deviation = √(variance)
Taking the square root of the variance brings the units back to those of the original data.
🔍 Why square?
①Adding all the deviations, the positive and negative ones cancel and the total is always 0.
②So the deviations are squared, making them all 0 or more, and then averaged.
③That average is the variance.
④The variance is in squared units (points²), which are hard to interpret.
⑤Its square root, back in the original units (points), is the standard deviation.
Worked examples
Example 1
Find the median of the data 1, 3, 5, 7, 9.
1
Put the data in order of size; they are already in order.
1, 3, 5, 7, 9
2
There are 5 values, an odd number, so the middle (third) value is the median.
median = 5
▸ 5
With an odd number of values the median is the middle one; with an even number it is the mean of the middle two.
Example 2
Find the variance of the data 1, 3, 5.
1
Find the mean: (1 + 3 + 5) ÷ 3 = 3.
mean = (1+3+5)/3 = 3
2
Square the deviations −2, 0 and 2 and take their mean.
variance = (4+0+4)/3 = 8/3
▸ 8/3
The variance is the mean of the squared deviations, where each deviation is a value minus the mean.
Exam checklist
Key facts about spread
V = Σ(dev)2n, σ = √V
The smaller the standard deviation, the more closely the data gather around the mean.
G9 school-exam type
The mean of the five values 5, 6, 8, 9, x is 7. Find x.
①5
②6
③7
④8
⑤9
▸ ③ 7
1
Mean = (sum) ÷ (count), so the sum = 7 × 5 = 35.
5+6+8+9+x = 35
2
28 + x = 35, so x = 7.
28+x=35 ⇒ x=7
🎯 Exam points
①The deviations always add up to 0; use this to check calculations.
②The variance is the mean of the squared deviations, and the standard deviation is its square root.
③A small standard deviation means the data are gathered evenly near the mean.
④When there are extreme values, the median is a better measure than the mean.
⑤The variance is in squared units; the standard deviation is in the original units.