What is a standard deviation?

Question

How do you work out the SD of a total?

Answer 1

In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. It is usually denoted with the letter σ (lower case sigma). It is defined as the square root of the variance.

To understand standard deviation, keep in mind that variance is the average of the squared differences between data points and the mean. Variance is tabulated in units squared. Standard deviation, being the square root of that quantity, therefore measures the spread of data about the mean, measured in the same units as the data.

Answer 2

Standard deviation is a measure of how good the mean average is relative to the individual elements in a set of data.

It is calculated by taking the difference between the mean average and each individual peice of data, squaring the difference, summing the squares, then dividing the sum by the number of peices of data. and then finally taking the square root.

This gives you an average difference (or deviation) between the mean average and each individual peice of data.

The higher the standard deviation, the worse the mean is as a representative value of the data set.

Its easiset to see how it works in some examples:

E.g (1) A student stops 10 middle aged couples in the high street and asks them how many children the have in their family. The results of the survey are as follows,

2,3,3,2,1,2,0,3,3,1

The total number of children = 2+3+3+2+1+2+0+3+3+1 = 20
The mean average number of children / couple = 20/10 = 2

The difference between each data element and the average is:

2-2 = 0, 3-2 = 1 , 3-2 = 1 , 2-2 = 0, 1-2 = -1, 2-2 = 0, 0-2 = -2,
3-2 = 1, 3-2 = 1, 1=2 = -1

The sqares of each of the differences above are:

0,1,1,0,1,0,4,1,1,1

Summing these gives 0+1+1+0+1+0+4+1+1+1 = 10

10 divided by number of elements (10) = 1

Square root of 1 = 1 therefore standard deviation = 1

Therefore the mean is moderately representative of the data set.

Eg.(2)

A vet surveys 10 cat owners who have female cats as to how many kittens each cat has had that survived past the first day.

The responses are as follows:

4,0,0,4,0,2,4,4,1,1

The sum again = 20 and there are 10 elements in the data set
therfore on average each cat had 2 kittens which survived.

The differences this time between the mean and the data are:

2,-2,-2,2,-2,0,2,2,-1,-1

Summing the sqares gives:

4+4+4+4+4+0+4+4+1+1 = 30

Dividing by the number of elements (10) gives 3.

The square root of 3 is 1.73 indicating that the mean average has an error of +/- 1.73 and that it is a poor indication of the data set.

Answer 3

Standard deviation is the square root of the variance. And variance is the sum of the difference of an individual data from the mean of the popoulation divided by the number of data points in the population. For sample variance and standard deviation, we use n-1 instead of n, where n is the number of data points in the sample.

A total itself cannot be used to calculate the variance and standard deviation. We need either the indivdiual data values or at least a set of grouped data.

Answer 4

First take the mean of the data set.
Say there are N data points and the mean is M
Call your data values x and let "i" be the index that identifies any given point. So x(i) is the ith data point.

The standard deviation is defined as:

SD = SQRT(SUM[x(i) - M)^2])/N

If we call x(i)-M the deviation of any point from the mean of the set of data points, then SD is the square root of the sum of the squares of the deviations for all points divided by the number of points.

Example:

x = 6,3,2,4,5
Number of data points = N = 5
Mean = M = SUM(x(i))/5 = 20/5 = 4

Sum of deviations = SUM[(x(i) - M)^2]
Sum of deviations = (6-4)^2 + (3-4)^2 + (2-4)^2 + (4-4)^2 + (5-4)^2
Sum of deviations =2^2 + (-1)^2 + (-2)^2 + 0^2 + 1^2
Sum of deviations = 4 + 1 + 4 + 1 = 10

SD = SQRT(Sum of deviations)/N = SQRT(10)/5