Comparing spread between groups

Science and society, grade 11 (3 units) Lesson 10 of 16Learning

Comparing spread between groups

The standard deviation from a frequency table

When the data are set out in a frequency table, each squared deviation is multiplied by its frequency – as in a weighted mean – and the sum is divided by the sum of the frequencies:

\displaystyle \sigma=\sqrt{\frac{f_1(x_1-\bar{x})^2+f_2(x_2-\bar{x})^2+\dots+f_k(x_k-\bar{x})^2}{f_1+f_2+\dots+f_k}}

The number of children in 20 families in a neighbourhood:

Number of children $\displaystyle x$	0	1	2	3	4	Sum
Families $\displaystyle f$	2	4	8	4	2	20
$\displaystyle x-\bar{x}$	$\displaystyle -2$	$\displaystyle -1$	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 2$
$\displaystyle f(x-\bar{x})^2$	8	4	0	4	8	24

The mean: $\displaystyle {\bar{x}}={\frac{2\cdot0+4\cdot1+8\cdot2+4\cdot3+2\cdot4}{20}}={\frac{40}{20}}={2}$ .
The standard deviation: $\displaystyle {\sigma}={\sqrt{\frac{24}{20}}}={\sqrt{1.2}\approx1.10}$ children.

A large and a small standard deviation

A small standard deviation says that most of the data are near the mean; a large one, that many data are far from it, above and below.

In two classes the mean mark in a test is 75.
In class A the standard deviation is 5, and in class B 15.
Each class has 30 pupils. The chart: how many pupils got a mark in each range.

In class A almost all the marks are between 60 and 89.
In class B there are marks below 50, and of 90 and above.
The mean alone does not tell the classes apart. The standard deviation does.

Choosing by the standard deviation

When two options give the same on average, the standard deviation says which of them is steadier – and sometimes that is what decides.

In a factory two machines fill 500 ml bottles.
Machine A: mean 500 ml, standard deviation 2 ml.
Machine B: mean 500 ml, standard deviation 8 ml.

On average both machines fill exactly 500 ml.
Machine B will leave many bottles short – and buyers will complain – and fill many too full, which costs the factory money.
The factory will prefer machine A, whose filling is steadier.

Adding to and multiplying all the data

When the same number is added to all the data, they all move together, and the spread does not change. When all the data are multiplied by a number, the distances between them are multiplied too.

Four marks in a quiz out of 50: 30, 30, 40, 40.
The mean is 35, and the standard deviation 5 (each mark is 5 from the mean).

The teacher adds 5 points to each mark: 35, 35, 45, 45. The mean rises to 40, and the standard deviation stays 5.
Turning them into marks out of 100 – multiplying by 2: 60, 60, 80, 80. The mean is 70, and the standard deviation doubles, to 10.

The box plot

A box plot shows in one picture where the middle of the data is and how spread out they are. So the spread of two groups is compared at a glance.

The journey to school, in minutes, of nine pupils in each of two classes:
Class A: 10, 12, 13, 14, 15, 15, 16, 17, 18.
Class B: 3, 6, 9, 12, 15, 18, 22, 25, 30.

The line inside the box is the median: 15 minutes in both classes.
The box holds the middle half of the data; the lines on either side of it reach the least and the greatest datum.
In class B the box and the lines are much wider: the journey times are more spread out.

Exercise 1

The marks of 10 pupils in a quiz out of 10:

Mark	6	7	8	9	10
Pupils	1	2	4	2	1

Work out the mean.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power