Variance and standard deviation

Worked example

These are the data on the weights of thirty cattle that we gave in the More Information page on Measures of location.

Using the calculator formula to calculate the variance we get:

s2 =     7629900  −   227406400 30 29

The sample standard deviation (s) is then the square root of the variance.

Standard deviation (s)   =   √1713.3 =   41.4

Worked example

Let us assume you have the following observations of bird weight (in grams), which have been divided into class intervals, but you not know their individual values, nor their mean.

Lacking their mean, we can estimate it by multiplying the mid-point of each class by the number of observations it contains.

Now we can work out the deviation of each mid-class from the sample mean. Then multiply the square of this deviation by the number of observations it refers to.

We then estimate the variance of this sample as

s² = 6474.2 / 45, or 143.9

Worked example

Let's take an example of packed cell volume values of five lambs:

Lamb No.	PCV
1	33
2	26
3	29
4	32
5	31
ΣY	151

The sample variance (s²) is calculated in the usual way:

s2   =    4591   −     22801    =   7.7 5 4

The (uncorrected) sample standard deviation (s) is then:

s   =   √ 7.7 =   2.775

Since we have a very small sample (n = 5), we need to correct this by multiplying by C_n.

The corrected standard deviation is then given by:

s_corr.    =    C_n × s    =    1.064 × 2.775    =   2.953

Worked example

Let's take an example of two repeated measurements of packed cell volume.

By taking the square root of the mean of the variances we get:
Within-subject standard deviation = √(28.5/10) = 1.688

Cow No.	PCV		si2	di2
	1.	2.
1	33	35	2	4
2	26	23	4.5	9
3	29	29	0	0
4	32	35	4.5	9
5	31	28	4.5	9
6	31	31	0	0
7	31	34	4.5	9
8	31	29	2	4
9	35	33	2	4
10	21	24	4.5	9
Sum			28.5	57

By taking the square root of half the mean of the differences we get:
Within-subject standard deviation = √(57/20) = 1.688

Hence, assuming the observations are normally distributed:

• the difference between a measured PCV and the true value should be less than (1.96 × 1.688) or 3.3 for 95% of observations.

• the difference between two PCV measurements on the same cow should be less than √2 × 1.96 × 1.688 or 4.68 for 95% of pairs of observations.