What is standard deviation?

Salters-Nuffield Advanced Biology Resources

Maths and Stats Support 10

STANDARD DEVIATION

What is standard deviation?

Standard deviation is a measure of the spread of the data about the mean value. It provides a measure of the `typical' amount that the values differ from the mean. Between them, the mean and standard deviation provide a good summary of a set of normally distributed data. (Normally distributed data are data that, if presented in a tally chart or plotted in a frequency histogram, form a symmetrical bellshaped graph.)

Using standard deviation

Alarm and despondency are spreading through the ranks of the NSPB (National Society for the Protection of Beards). The trustees of the organisation are deeply concerned that stubbly `designer beards' have been observed infecting the chins of some members. These `shorty' appendages are strictly illegal (minimum permissible beard length of 0.5 m and a willingness to house a homeless badger being the basic requirements for members). You have been employed to investigate suspect groups and report back to the Minimum Standards (pogonophilia) Committee.

You collect the data shown in Table 1.

Membership category Beard lengths/metres

Biology professors

0.1 0.9 0.4 0.5 0.5 0.5 0.5 0.6 0.5 0.5

Zoo keepers

0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.6

Eco-warriors

0.1 0.2 0.5 0.8 0.5 0.7 0.4 0.5 0.6 0.7

Table 1 Maximum beard lengths for 10 individuals in each of three categories of members of the NSPB.

You can plot these data as line plots and also work out the mean length for each category.

Figure 1 Line plots of beard length amongst NSPB members.

Notice that the mean beard length of all groups is the same, 0.5 m, the minimum permissible NSPB beard length. However, the data for each group are different in the way that the data points are scattered about the mean.

All users will need to review the risk assessment information and may need to adapt it to local circumstances.

? 2015 University of York, developed by University of York Science Education Group. This sheet may have been altered from the original.

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Salters-Nuffield Advanced Biology Resources

Maths and Stats Support 10

How can we describe the differences in the three datasets? What is needed is a measure that indicates the spread of the data about the mean. The standard deviation is just such a measure.

Calculating standard deviations

Standard deviation of biology professors' beard lengths in easy steps

1 Set out the raw data in the first column of a table (Table 2) and then work out the mean, x . The mean beard length of the biology professors is 0.5 m.

2 Work out the deviation of each piece of data from the mean (x ? x ), the value x in column one minus the mean. If we just added the second column up the values would cancel each other out and we would end up with a value of zero. Obviously this is no use as a measure of scatter of the data about the mean. One way around this problem is to square each of the values in the second column. The negative values now become positive values.

3 Square each of the deviations (x ? x )2. 4 Add up the squared deviations (x ? x )2. This has been done at the bottom of the third column;

the sum of the squares is 0.34.

Biology professors' beard (x ? x ) length, x

0.1

?0.4

0.9

0.4

0.4

?0.1

0.5

0.0

0.5

0.0

0.5

0.0

0.5

0.0

0.6

0.1

0.5

0.0

0.5

0.0

x = 0.5

Table 2 Working out the sum of the squared deviations.

(x ? x )2

0.16 0.16 0.01 0.0 0.0 0.0 0.0 0.01 0.0 0.0 (x ? x )2 = 0.34

5 Divide the sum of the squares, value 0.34 by (n ? 1), the degrees of freedom.

(x x)

n 1

=

0 .34 9

= 0.038

We now have a number, 0.038, that represents the scatter or spread of our biology professors' beard lengths about the mean value. Statisticians call this number the variance (s2) of the data and

it is used in numerous ways (see Maths and Stats Support Sheet 11 ? t-tests ? for one example).

All users will need to review the risk assessment information and may need to adapt it to local circumstances.

? 2015 University of York, developed by University of York Science Education Group. This sheet may have been altered from the original.

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Salters-Nuffield Advanced Biology Resources

Maths and Stats Support 10

6 To complete our calculation of standard deviation we must take the square root of the variance to convert the number back to its original units, in this case length in metres. (Remember we squared all our differences from the mean to get rid of negative values.)

The standard deviation (s) of our sample of biology professors' beard lengths is

variance = 0.038 = 0.194 m.

The calculation we have just done can be represented by this formula:

s = (x x)2 n 1

If we do the calculations for the zoo keepers and the eco-warriors we get the following values:

Biology professors: s = 0.194 m

Zoo keepers:

s = 0.047 m

Eco-warriors:

s = 0.221 m.

Notice that the zoo keepers (whose beards were all clustered around the mean) have a very small standard deviation. The biology professors (whose beards were mostly clustered around the mean

but not to the same extent as the zoo keepers) have a bigger value and the eco-warriors, whose beard lengths were all over the place (i.e. widely scattered about the mean), have the biggest

standard deviation of all.

Most scientific calculators or spreadsheets will save you the tedium of the above calculations if you know how to operate their statistical functions. The instructions will tell you how to do this. Alternatively, you can use the formula below that is a bit quicker (mathematically it's the same as

the one used above).

s =

x 2

( x)2 n

n 1

Where: x2 = each individual piece of data squared and then added up

(x)2

= each individual piece of data added up and then squared

n

= the number of items of data.

What does the standard deviation tell us?

A small standard deviation tells us that the data are clustered around the mean; a large standard deviation tells us that the data are more spread out around the mean. Normally about 65% of the individual values of the data lie within one standard deviation of the mean and about 95% lie within two standard deviations of the mean. Using the standard deviation we can work out standard error. The standard error provides an estimate of how close the sample mean probably is to the mean for the whole population (in our example the population is the mean beard length of all members of the NSPB). We can work out the standard error using the formula:

se = s n

For biology professors the standard error would be:

se = 0.194 m 3.162

= 0.061 m Standard errors are often presented on graphs as vertical lines above and below the mean value, providing a measure of the precision of the mean.

All users will need to review the risk assessment information and may need to adapt it to local circumstances.

? 2015 University of York, developed by University of York Science Education Group. This sheet may have been altered from the original.

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