The Sample The Sample VarianceVariance

© Chistine CrispEdited by Dr Mike Hughes

The Sample Variance

Can you find the medians and means for the following 3 data sets?

Although the medians and means are the same, the data sets are not really alike.

The spread or variability of the numbers is quite different.How can we measure the spread within the

data sets?ANS: The range and inter-quartile range both measure spread but neither uses all the data items.

5

5

5 5

55

955555551Set C

999654111Set B

987654321Set A

Mean,Median

x

Interquartile range we will do later with Cumulative Frequency

The Sample Variance

If you had to invent a method of measuring spread that used all the data items, what could

you do?One thing we could do is find out how far each item is from the mean and add up these differences.

e.g.

)( xx 4 3 . . . + 3 + 4 =Data sets B and C give the same result. The negative and positive values have cancelled each other out.

432101234

55955555551Set C

55999654111Set B

55987654321Set A

Mean,Median

x

987654321Set A: xxx

5x

0

The Sample Variance

To avoid the effect of the negative values we can either • ignore the negative

signs, or• square each difference ( since the squares will all be positive ).

Squaring is more convenient for developing theory, so, e.g.

16941014916

432101234987654321Set A: x

xx 2)( xx

2)( xx 60

Let’s do this calculation for all 3 data sets:

The Sample Variance

98 3260 2)( xx 2)( xx 2)( xxSet A: Set B: Set C:

The larger value for set B shows greater variability. Set C has least variability.Can you see a snag with this

measurement?ANS: The calculated value increases if we have more data, so comparing data sets with different numbers of items would not be possible.

To allow for this, we need to take n, the number of items, into account.

5955555551Set C: x

5999654111Set B: x

5987654321Set A: xMean, x

The Sample Variance

There are 2 formulae that can be used,

n

xxmsd

2)( the mean square

deviation.

1

)( 22

n

xxs

or

the sample variance.

Our data is nearly always a sample from a large unknown set of data ( the population ) and we take samples to find out about the population. The 1st formula does not give the best estimate of the variance of the population so is not used.

The Sample Variance

So, there are 2 quantities and their square roots that we need to be clear about

n

xxmsd

2)( the mean square

deviation,POPULATION VARIANCE

1

)( 22

n

xxs

Also

the sample variance,

and

n

xxrmsd

2)(

the root mean square deviation.POPULATION STANDARD DEVIATION

and

1

)( 2

n

xxs the sample

standard deviation.

WE nearly ALWAYS use THESE TWO

formula

The Sample Variance

e.g. Find the rmsd and msd of the following data:

n

xx 22 )( (i

)

x 7 9 14

Mean, n

xx

103

30 x

3

)1014()109()107( 222

)..3(6783

1619fs

22

2 xn

x(ii) 210

3

1968149

100

3

326

)..3(678 fsThe 2nd form is exactly the same as the first form but quicker to use !!

The Sample Variance

e.g. Find the sample SD and Variance of the following data:

1

)( 22

n

xxs(i

)

x 7 9 14

Mean, n

xx

103

30 x

2

)1014()109()107( 222

)..3(0.132

1619fs

1

222

n

xnxs(ii)

2

1031968149 2 150

2

326

)..3(0.13 fs

The 2nd form is in general quicker to use.

The Sample Variance

This all seems very complicated but help is at hand.

,942)( 2

n

xxrmsd 613

1

)( 2

n

xxs

Both the quantities, rmsd and s are given by your calculator.

The rmsd is smaller than s ( because we are dividing by a larger number ). Correct to 3 s.f. we have

e.g. Find the root mean square deviation, rmsd, and the sample standard deviation, s, for the following data: x 7 9 14

Use the Statistics function on your calculator and enter the data. Select the list of calculations.You will be able to find the following: ...605553 ...943922 and

The Sample Variance

,942)( 2

n

xxrmsd 613

1

)( 2

n

xxs

x 7 9 14So, for the data

we have

Squaring these gives

,678)( 2

n

xxmsd 13

1

)( 22

n

xxs

( sample variance )

( mean square deviation )

The part of the formula, , is in your

formulae sheet, labelled Sxx. (said as Sum of

squares X X)

2)( xx

An expanded form of the expression is also given. All you have to do is divide by the correct quantity.

The Sample Variance

The mean square deviation, msd, and sample variance, both measure the spread or variability in the data.

SUMMARY

To find the msd or sample variance, we square the relevant quantity given by the calculator:

If we have raw data we use the statistical functions on the calculator to find the rmsd or sample standard deviation.

msd = (rmsd)2 sample variance s2 Your formulae sheet will gives the formula or

equivalent: n

xxxxS xx

2

22)(

Then, we divide by n for the msd or (n – 1) for s2.

The sample standard deviation is the larger than the rmsd because we divide by (n-1)

The Sample Variance

The formula for the variance can be easily adapted to find the variance of frequency data.

n

xxxxS xx

2

22)(

Becomes for FREQUENCY DATA

Frequency Data

f

xffxxxfS xx

2

22)(

We usually only use the formulae if we are given summary data. With raw data we enter the data into the calculator and use the statistical functions to get the answers directly.

The Sample Variance

But note that ...............

22

2

22)( xnxn

xxxxS xx

becomes

Frequency Data

22

2

22)( xnfxf

xffxxxfS xx

The Sample Variance

SO MSD= SXX/n and VARIANCE = SXX/(n-1)

2222

xn

x

n

xnxmsd

becomes

Frequency Data

11

2222

n

xnx

n

xnxVariance

2222

xn

fx

n

xnfxmsd

11

2222

n

xnfx

n

xnfxVariance

The Sample Variance

e.g.1 Find the mean and sample standard deviation of the following data:

x 1 2 5 10

Frequency, f

3 5 8 4

Solution:

654

sample standard deviation,

)..3(173 fss

Using the calculator functions, the mean, =

Although we don’t need the formula for this question, let’s check we have the correct value by using the formula:

The Sample Variance

e.g.1 Find the mean and sample standard deviation of the following data:

x 1 2 5 10

Frequency, f

3 5 8 4

Solution:

f

xffxS xx

2

2

20

)410...31(410...31

222

xxSSo,

55190

12

n

Ss xx )..3(173 fss 02910

19

55190

The Sample Variance

Length (cm)

1-9 10-14 15-19 20-29

Frequency, f

2 7 12 9

e.g.2 Find the sample standard deviation of the following lengths:

The Sample Variance

e.g.2 Find the sample standard deviation of the following lengths:

Length (cm)

1-9 10-14 15-19 20-29

x

Frequency, f

2 7 12 9

Solution:

Standard deviation, s =

)..3(775 fs

We need the class mid-values

5 12 17 24·5

x2 25 144 289 600.25

x2f 50 1008 3468 5402.25

xf 10 84 204 220.5 283.17

f

xfx

1

22

n

xnfxVarianceSample 351.33

29

)283.17(3025.9928 2

30 fn

The Sample Variance

e.g.3 Find the mean and sample variance of 20 values of x given the following:

Solution:

82x 3702 xand

n

xx

sample mean,

1420

82x

Since we only have summary data, we must use the formulae

sample variance,

12

n

Ss xx

22 xnxS xx )1.4(20370 2xxS

833

)..3(781 fs

The Sample Variance

SUMMARY

1var

22

n

xnfxianceSample

1

22

n

xnxVarianceSample

n

xnxmsd

22

n

xnfxmsd

22

Frequency data

Raw data

MSD is called POPULATION VARIANCE

Take square root for rmsd and sample standard deviation

RMSD is called POPULATION STANDARD DEVIATION

The Sample Variance

ExerciseFind the mean, sample standard deviation and sample variance for each of the following samples, using calculator functions where appropriate.

1. 8121497f

54321x

2.

8121497f

21-2516-2011-156-101-5Time ( mins )

3. 10 observations where and432 x 189122 x

The Sample Variance

1. 8121497f

54321x

mean, 13x

variance, 6412 s

standard deviation, s = )..3(281 fsAnswer

:

Answer:

mean, 513x

variance, )..3(1412 fss standard deviation, s = )s.f.( 3416

23181383x

Time ( mins )

8121497

2. 21-2516-2011-156-101-5

f

N.B. To find we need to use the full calculator value for s, not the answer to 3 s.f.

2s

The Sample Variance

3. 10 observations where and432 x 189122 x

Solution:

Standard deviation, s 727

243 xn

xxmean

,

variance,1

2

n

Ss xx

)s.f.( 3275

) s.f. (3 7272 s

n

xxS xx

2

2 10

)432(18912

2

xxS

6249

The Sample Variance

n

xxmsd

2)(

There are 2 formulae that can be used to measure spread:

1

)( 22

n

xxs

or

the mean square deviation.

the sample variance,

In many books you will find the word variance used for the 1st of these formulae and you may have used it at GCSE.However, our data is nearly always a sample from a large unknown set of data ( the population ) and we take the sample to find out about the population. The 1st formula does not give the best estimate of the variance of the population so is not used.

The Sample Variance

n

xxmsd

2)(

So, there are 2 quantities and their square roots that we need to be clear about

1

)( 22

n

xxs

Also

the mean square deviation

the sample variance,

and

n

xxrmsd

2)( the root mean

square deviation.

and

1

)( 2

n

xxs the sample

standard deviation.

The Sample Variance

,942)( 2

n

xxrmsd 613

1

)( 2

n

xxs

The rmsd is smaller than s ( because we are dividing by a larger number ). Correct to 3 s.f. we have

e.g. Find the root mean square deviation, rmsd, and the sample standard deviation, s, for the following data:

1497x

Use the Statistics function on your calculator and enter the data. Select the list of calculations.You will be able to find the following: ...605553 ...943922 Ignore the calculator notation.

The Sample Variance

Squaring these gives

,678)( 2

n

xxmsd 13

1

)( 22

n

xxs

( variance )

( mean square deviation )

The part of the formula, , is in your

formulae booklet ( see correlation and

regression ), labelled Sxx.

2)( xx

An expanded form of the expression is also given. All you have to do is divide by the correct quantity, n or n 1.

Using the formulae:If summary data are given, you will need to use the formulae instead of the calculator functions.

The Sample Variance

The mean square deviation, msd, and sample variance, both measure the spread or variability in the data.

SUMMARY

To find the msd or sample variance, we square the relevant quantity given by the calculator:

If we have raw data we use the stats functions on the calculator to find the rmsd or sample standard deviation.

msd = (rmsd)2 sample variance s2 For summary data, we use the formulae book,

choosing the appropriate form: n

xxxxS xx

2

22)(

Then, we divide by n for the msd or (n – 1) for s2.

The sample standard deviation is the larger of these quantities.

The Sample Variance

e.g.1 For the following sample data, find (a) the root mean square deviation, rmsd,(b) the mean square deviation, msd, (c) the sample standard deviation, s, and(d) the sample variance s2.

9141512x

Answer: Using the calculator functions,rmsd(a

)msd(b

)s(c) 2s(d

)

)..3(292 fs 255

)..3(652 fs 7

The Sample Variance

e.g.2 Given the following summary of data for a sample of size 5, find

24)( 2 xx,5n

Solution: Using the formulae book,

n

S xx

64

242 s

)..3(4526 fss

(a) the mean square deviation, msd, (b) the root mean square deviation,

rmsd ,(c) the sample variance s2

(d) the sample standard deviation, s , and,

msd =(a)

(b)

12

n

Ss xx

(c)

(d)

2)( xxS xx

845

24

rmsd = )..3(19284 fs

The Sample Variance

The formula for the variance can be easily adapted to find the variance of frequency data.

n

xxxxS xx

2

22)(

becomes

Frequency Data

f

xffxxxfS xx

2

22)(

As before, we only use the formulae if we are given summary data.

The Sample Variance

e.g.1 Find the mean and sample standard deviation of the following data:

4853Frequency, f

10521x

Solution:

f

xffxxxfS xx

2

22)(

20

410...31410...31 22 xxSSo,

55190

0391019

55190

12

n

Ss xx )..3(173 fss

The Sample Variance

e.g.2 Find the sample standard deviation of the following lengths:

91272Frequency, f

20-2915-1910-141-9Length (cm)

Solution:

Standard deviation, s =

)..3(775 fs

We need the class mid-values

5 12 17 24·5

We can now enter the values of x and f on our calculators.

x

91272Frequency, f

The Sample Variance

To find the root mean square deviation, rmsd, or the sample standard deviation, s, using the calculator functions,

SUMMARY

• the values of x ( and f ) are entered and checked,

• the table of calculations gives both values,

• the variance is the square of the standard deviation.

• the larger value is the sample standard deviation, s, and this is the value that is most often used by statisticians,

The Sample Variance

Outliers

We’ve already seen that an outlier is a data item that lies well away from the other data. It may be a genuine observation or an error in the data.

e.g. 1 Consider the following data: 81211917141210

With this data set, we would immediately suspect an error. The value 81 was likely to have been 18. If so, there would be a large effect on the mean and standard deviation although the median would not be affected and there would be little effect on the IQR. The presence of possible outliers is an argument in favour of using median and IQR as measures of data.

The Sample Variance

A 2nd method used to identify outliers is to find points that are further than 2 standard deviations from the mean.

The point 33 is more than 2 standard deviations above the mean so, using this measure, it is an outlier.

In an earlier section, we met a method of identifying outliers using a measure of 1·5 IQR above or below the median.

e.g. 2. Consider the following sample:

21 22 24 33191817141210

The sample mean and sample standard deviation are : mean, 19x

standard deviation, s = )..3(626 fs2132 sSo,232213 xand