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Measures of Variability

For a given set of data, we might want to know how much data changes for a sample or a population. There are different ways of expressing this change in the data. The three common ways are by means of the range, variance, and standard deviation.

Range

The range of a sample of population can be described as the difference between the maximum and minimum of a given sample or population. In other words

Range= Max(X)-Min(X)

The range is useful as it helps to show how variable data might be. One of the ways this is useful is that it gives us an idea of how much a thing changes in comparison to another thing. For example, the range for a set of experiments calculating gravity will be much smaller than the range of the number of households in a town throughout the United States.

Variance

The variance describes how different the sample or population is from the mean. This is valuable to us because it gives us an idea if individuals vary greatly or very little. Variance for the population can be expressed mathematically in the following way

\sigma^2= \frac{\sum_{i=1}^{N} y_i - \mu}{N}

Variance for the sample can be expressed mathematically in the following way

s^2= \frac{\sum_{i=1}^n y_i - \bar{y}}{n-1}

What is essentially happening for the sample variance is

  • For each observation, subtract it by the mean
  • Square the difference
  • Sum all the squared differences together
  • Divide the sum by the number of observations minus 1
Example I

Suppose that we have the data described in Table 1. Suppose that this data came from an experiment where we calculated the force of gravity 5 times. Calculate the variance.

Table 1
Trial Observed Value
1 9.70
2 9.85
3 9.77
4 9.72
5 9.76

 

First we need to calculate the mean of the observed values. If you do not know what the mean is, click here to review that topic. Otherwise, apply the following equation were we sum up all the observations and then divide by the number of observations.

\bar{x}=\sum_{i=1}^n \frac{x_i}{n}

\bar{x}= \frac{9.70+9.85+9.77+9.72+9.76}{5}

\bar{x}=\frac{48.82}{5}

\bar{x}=9.76

Now that we have the sample mean, we will calculate the sample variance.

s^2= \frac{\sum_{i=1}^n (y_i - \bar{y})^2}{n-1}

s^2= \frac{{(9.70-9.76)^2}+{(9.85-9.76)^2}+{(9.76-9.76)^2}+{(9.72-9.76)^2}+{(9.77-9.76)^2}}{5-1}

s^2= \frac{{(-0.06)^2}+{(0.09)^2}+{(0.01)^2}+{(-0.04)^2}+{(0.00)^2}}{5-1}

s^2= \frac{0.0036+0.0081+0.0001+0.0016+0.0000}{4}

s^2= \frac{0.0134}{4}

s^2= 0.00335

 

Standard Deviation

The standard deviation described how much the average observation differs from the typical observation. The standard deviation is more useful for a variety of reasons, and it is used heavily in elementary statistics. We will see some of its applications in the next section.

The population standard deviation can be described mathematically as

\sigma= \sqrt{\frac{\sum_{i=1}^{N} y_i - \mu}{N}}

The sample standard deviation can be described mathematically as

s= \sqrt{\frac{\sum_{i=1}^{n} y_i - \bar {y}}{n-1}}