Year 10 Extended · Unit 6 Statistics and the bell curve · Lesson 3 of 7

One number for how far values typically stray

Two data sets can share a mean and still be nothing alike. This lesson builds one number for how widely values are scattered, and gives a reason for every step in it.

Statistics and ProbabilityGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 8: find the mean of 3, 7, 8 and 10.
    Answer7. The total is 28 and 28 ÷ 4 = 7.
  2. From Year 9: work out (−5)².
    Answer25. A negative number squared is positive: −5 × −5 = 25.
  3. From Year 9: what is the range of 14, 22, 9, 31 and 18?
    Answer22. The greatest is 31, the least is 9, and 31 − 9 = 22.
Words used in this lesson
deviation
the signed distance of a value from the mean. For example: x − μ = −1.5.
variance
the mean of the squared deviations. For example: σ² = 2.25.
standard deviation
the square root of the variance, a typical distance from the mean. For example: σ = 1.5.
spread
how widely values are scattered. For example: the same mean, a different spread.
population
the whole set of values being described. For example: every pupil in the year group.
root mean square
the square root of the mean of the squares. For example: √(Σd² ÷ n).

Two towns with the same July

Two towns on the same coast both have a mean July afternoon temperature of 20°C. In one town every afternoon is close to 20. In the other the afternoons swing from cold to hot. You are planning an outdoor festival.

Predict first: how could you capture “how much the temperature varies” in one number that is better than the range? Then use the figure to compare four data sets that share a mean.

1
20mean
2mean of the squares of the deviations
1.41standard deviation

Data set 1: 18, 19, 20, 21, 22. The mean is 20, the mean of the squared deviations is 2, and the standard deviation is 1.41.

Show me the steps
  1. Keep set 1. The values are 2 or less from the mean 20. The squared deviations are 4, 1, 0, 1, 4, so the mean square is 10 ÷ 5 = 2 and the standard deviation is √2 = 1.41.
  2. Choose set 2. The mean is still 20, but the deviations are −10, −5, 0, 5, 10. The squares are 100, 25, 0, 25, 100, the mean square is 50, and the standard deviation is 7.07, five times as large.
  3. Choose set 3. Its range is 16, twice set 4's range of 8, but its standard deviation, 5.06, is only about 1.4 times set 4's 3.58: three of its five values sit on the mean, and only two are far away.
  4. Choose set 4. The deviations are −4, −4, 0, 4, 4, the mean square is 12.8 and the standard deviation is 3.58. All four sets have mean 20; only the deviations tell them apart.

The deviation of a value is x − μ, its signed distance from the mean μ. The deviations of any data set add to zero, so they cannot be averaged as they are. Squaring makes every one positive.

The standard deviation σ is the root of the mean of the squared deviations: σ = √(Σ(x − μ)² ÷ n). The mean of the squares is the variance, σ². Taking the root brings the units back to those of the data.

A small σ means values stay close to the mean; a large σ means they are scattered. It is a measure of spread that uses every value, unlike the range, which uses only two.

You cannot average the deviations themselves. For 3, 5 and 10 the mean is 6 and the deviations are −3, −1 and 4, which add to 0. An average of 0 would claim there is no spread at all, although the values are different. Squaring each deviation first stops the positives and negatives cancelling.

Worked examples

1. Find the standard deviation of 7, 9, 10, 12 and 12.

  1. The mean is 50 ÷ 5 = 10.
  2. The deviations are −3, −1, 0, 2, 2, and their squares are 9, 1, 0, 4, 4, which add to 18.
  3. The variance is 18 ÷ 5 = 3.6, and σ = √3.6 = 1.9 to one decimal place.

2. Why do the deviations always add to zero?

  1. Σ(x − μ) = Σx − nμ, because μ is subtracted n times.
  2. But μ = Σx ÷ n, so nμ = Σx, and the sum is Σx − Σx = 0.

3. Set P is 4, 6, 8 and set Q is 2, 6, 10. Compare their standard deviations.

  1. Both have mean 6. For P the squares are 4, 0, 4 and σ² = 8 ÷ 3. For Q they are 16, 0, 16 and σ² = 32 ÷ 3.
  2. σQ² is 4 times σP², so σQ is 2 times σP. Q's deviations are each twice P's.

Your go. The first step is given. Find σ for 2, 3, 6 and 9, to one decimal place.

  1. The mean is 20 ÷ 4 = 5.
  2. Now find each deviation, square it, add, divide by 4, and take the root.
Check the answer

The deviations are −3, −2, 1 and 4, with squares 9, 4, 1 and 16. They add to 30, so the variance is 30 ÷ 4 = 7.5 and σ = √7.5 = 2.7.

With a GDC (optional)

Enter the data in a list and ask for the one-variable statistics. The display shows the mean and two standard deviations. In this lesson the data are the whole population, so read the one that divides by n (often written with the Greek letter sigma, and labelled as the population value), not the one that divides by n − 1.

Check one answer by hand against the display, so you know which of the two you are reading. Everything here can be done by hand.

Check before you go on

1. Five values have deviations from their mean of −3, −1, 0, 1 and 3. What is the standard deviation?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. For the data 2, 4, 6, 8, 10 the mean is 6. What is the mean of the squared deviations (the variance)?

3. For 8 values, Σ(x − μ)² = 72. What is the standard deviation?

Level 2 · Building

4. Find the standard deviation of 4, 8, 8 and 12, to one decimal place.

5. The data 2, 7, 7, 12 have mean 7. A fifth value, 7, is added. What is the standard deviation of the five values, to one decimal place?

Level 3 · Stretch

6. 10 values have mean 15 and standard deviation 3. What is Σ(x − μ)²?

7. Five values are 6, 6, 6, 6 and x, and their mean is 8. Find the standard deviation.

Something to think about

Could a data set have a standard deviation of exactly 0? Describe all such data sets. Then: if you add one more value that is equal to the mean to any data set, does the standard deviation go up, go down or stay the same, and why?

One way to think about it

A standard deviation of 0 means every deviation is 0, so every value equals the mean: all the values are the same. Adding a value equal to the mean adds a 0 to the sum of squares but makes n larger, so the mean square falls and so does the standard deviation, unless it was already 0.

Exit ticket

8. Each data set has 5 values and mean 10. Which has the greatest standard deviation?