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Averages/Dispersion-Solutions

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Question 1

Find the mean and standard deviation of the following numbers.

5,9,3,3,4,5,6,7,9,8

First we need to find the sum of the numbers

Therefore, since there are 10 numbers, the mean is

To find the standard deviation, we will need to square all of the numbers

x² = 25,81,9,9,16,25,36,59,81,64

and then find the sum

Therefore the standard deviation is

Substitute the numbers into the formula.

Write over a common denominator.

Square root to find the standard deviation.

Therefore

and

Question 2

A supermarket wanted to check the consistency of the weights of squashes it was selling. They therefore weighed a sample, putting the results in the table below. Find the mean weight and the standard deviation of the weights.

Mass (kg)
Number	4	10	22	12	2

To calculate the mean and standard deviation of this data, we need to draw up a table, including the midpoint, the midpoint squared and both of these multiplied by the frequency.

Mass (kg)	Frequency	Midpoint (m)	m²	fm	fm²
	4	1.3	1.69	5.2	6.76
	10	1.5	2.25	15	22.5
	22	1.7	2.89	37.4	63.58
	12	1.9	3.61	22.8	43.32
	2	2.1	4.41	4.2	8.82

84.6

144.98

We can now calculate the mean and the standard deviation, using the formulæ.

		Substitute in the sums from the table.
Therefore

and for the standard deviation

		Substitute in the sums from the table. Tidy up. Square root for standard deviation.
Therefore

Question 3

Find the mean and standard deviation of the following data.

x
f	6	17	30	27	12	8

We first need to put together a table, containing the midpoints and the various sums that are required to calculate the mean and standard deviation.

x	f	m	m²	fm	fm²
	6	2.5	6.25	15	37.5
	17	7	49	119	833
	30	11	121	330	3630
	27	14	196	378	5292
	12	18	324	216	3888
	8	23	529	184	4232

100

1242

17912.5

We now just need to put the sums into the formulæ

First the mean
Therefore the mean is

Now the standard deviation		Substitute in the values from the table. Tidy up. Square root to find the standard deviation.
Therefore the standard deviation is

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