Measures of Central Tendency

Measures of Central Tendency

With grouped data our estimates of central tendency will change because we lose some information when we place each value in a range. If all we have to work with is the grouped data, we do not know the measured values to the same accuracy as before. The best we can do is to assume that values are grouped at the centre of each range.

Looking back to the previous worked example, we started with this data set of learners' heights.

\begin{align*} \big\{132; 132; 132; 133; 138; 139; 139; 140; 141; 142; 142; 146; 150; 150; 152; \\ 152; 155; 156; 157; 160; 161; 162; 163; 164; 168; 168; 169; 169; 170; 172 \big\} \end{align*}

Note that the data are sorted.

The mean of these data is \(\text{151.8}\) and the median is \(\text{152}\). The mode is \(\text{132}\), but remember that there are problems with computing the mode of continuous quantitative data.

After grouping the data, we now have the data set shown below. Note that each value is placed at the centre of its range and that the number of times that each value is repeated corresponds exactly to the counts in each range.

\begin{align*} \big\{ 135; 135; 135; 135; 135; 135; 135; 145; 145; 145; 145; 145; 155; 155; 155; \\ 155; 155; 155; 155; 165; 165; 165; 165; 165; 165; 165; 165; 165; 175; 175 \big\} \end{align*}

The grouping changes the measures of central tendency since each datum is treated as if it occurred at the centre of the range in which it was placed.

The mean is now \(\text{153}\), the median \(\text{155}\) and the mode is \(\text{165}\). This is actually a better estimate of the mode, since the grouping showed in which range the learners' heights were clustered.