Interpretation and Application

Interpretation and Application

A large standard deviation indicates that the data values are far from the mean and a small standard deviation indicates that they are clustered closely around the mean.

For example, consider the following three data sets: \begin{align*} & \{65; 75; 73; 50; 60; 64; 69; 62; 67; 85\} \\ & \{85; 79; 57; 39; 45; 71; 67; 87; 91; 49\} \\ & \{43; 51; 53; \text{110}; 50; 48; 87; 69; 68; 91\}\end{align*}

Each of these data sets has the same mean, namely \(\text{67}\). However, they have different standard deviations, namely \(\text{8.97}\), \(\text{17.75}\) and \(\text{21.23}\). The following figures show plots of the data sets with the mean and standard deviation indicated on each. You can see how the standard deviation is larger when the data are more spread out.

\(\begin{array}{l@{\quad}r@{\;}l} \text{data:} & \{x_i\} &= \{65; 75; 73; 50; 60; 64; 69; 62; 67; 85\} \\ \text{mean:} & \overline{x} & = 67 \\ \text{standard deviation:} & \sigma &\approx \text{8.97}\end{array}\)

\(\begin{array}{l@{\quad}r@{\;}l} \text{data:} & \{x_i\} &= \{85; 79; 57; 39; 45; 71; 67; 87; 91; 49\} \\ \text{mean:} & \overline{x} & = 67 \\ \text{standard deviation:} & \sigma &\approx \text{17.75}\end{array}\)

\(\begin{array}{l@{\quad}r@{\;}l} \text{data:} & \{x_i\} &= \{43; 51; 53; \text{110}; 50; 48; 87; 69; 68; 91\} \\ \text{mean:} & \overline{x} & = 67 \\ \text{standard deviation:} & \sigma &\approx \text{21.23}\end{array}\)

The standard deviation may also be thought of as a measure of uncertainty. In the physical sciences, for example, the reported standard deviation of a group of repeated measurements represents the precision of those measurements. When deciding whether measurements agree with a theoretical prediction, the standard deviation of those measurements is very important: if the mean of the measurements is too far away from the prediction (with the distance measured in standard deviations), then we consider the measurements as contradicting the prediction. This makes sense since they fall outside the range of values that could reasonably be expected to occur if the prediction were correct.