Measures of Central Tendency - SS2 Mathematics Lesson Note

Measures of central tendency are statistical measures that evaluate broad trends across a data from its middle portion. The three main measures of central tendency include: mean, median and mode.

The mean (arithmetic mean) is sum of a group of numbers divided by the number of items in the group. Where each is a data item and f its frequency, the mean is calculated as \(\overline{X} = \frac{\sum_{}^{}{Xf}}{\sum_{}^{}f}\)

The median is the middle data item in a set when arranged either in ascending or descending order. For discrete quantities, the median is calculated as \(\frac{n + 1}{2},\ where\ n\ is\ the\ total\ number\ of\ items\ in\ the\ group\ or\ \sum_{}^{}f\). For grouped data, the median is \(l_{m} + (\frac{\frac{\sum_{}^{}f}{2} - {cf}_{cb}}{f_{m}})c\), where \(l_{m}\) is the lower class boundary of the median class, \(\sum_{}^{}f\) is the summation of the frequencies, \({cf}_{cb}\) is the cumulative frequency before the median class, \(f_{m}\) is the frequency of the median class and \(c\) is the class width

The mode is the item that appears most frequently in the group, that is, the item with the highest frequency.

Example: In the data below, what is the mean, median and mode of the data?

\[\mathbf{Scores\ in\ a\ physics\ test\ (out\ of\ a\ 100}\mathbf{)}\]	\[\mathbf{Number\ of\ students\ (}\mathbf{f}\mathbf{)}\]
\[100\ –\ 81\]	\[67\]
\[80\ –\ 61\]	\[34\]
\[60\ –\ 41\]	\[56\]
\[40\ –\ 21\]	\[12\]

Solution

\[Scores\]	\[f\]	\[Cumulative\]

\(Frequency\)

\[class\ mark\ (average\ of\ the\ class\ limits)\ X\] \[Xf\] \[100\ –\ 81\]\[67\]\[67\]\[90.5\]\[\ = PRODUCT(LEFT)\ 6063.5\]\[80\ –\ 61\]\[34\]\[67 + 34 = 101\]\[70.5\]\[\ = PRODUCT(LEFT)\ 2397.0\]\[60\ –\ 41\]\[56\]\[101 + 56 = 157\]\[50.5\]\[\ = PRODUCT(LEFT)\ 2828.0\]\[40\ –\ 21\]\[12\]\[157 + 12 = 169\]\[30.5\]\[\ = PRODUCT(LEFT)\ 366.0\] \[\sum_{}^{}f = \ = SUM(ABOVE)\ 169\] \[\sum_{}^{}{Xf} = \ = SUM(ABOVE)\ 11654.5\]

Note, Cumulative frequency is the sum of successive frequencies

\(Mean,\ \overline{X} = \frac{\sum_{}^{}{Xf}}{\sum_{}^{}f} = \ \frac{\ = SUM(ABOVE)\ 11654.5}{169} \approx 69\)

\(\frac{n + 1}{2} = \ \frac{169 + 1}{2} = \frac{170}{2} = 85\)

This means the median is 85^th item in the second class (\(80 - 61\)) [check the cumulative frequency], so we take the calculate the median for this grouped data as \(l_{m} + {(\frac{\frac{\sum_{}^{}f}{2} - {cf}_{cb}}{f_{m}})}^{c}\)

\[l_{m} = 60.5\]

\[\sum_{}^{}f = \ 169\]

\[{cf}_{cb}\ = 67\]

\[f_{m} = 34\]

\[c = 20\]

\[median = l_{m} + (\frac{\frac{\sum_{}^{}f}{2} - {cf}_{cb}}{f_{m}})c\]

\[= 60.5 + \left( \frac{\frac{169}{2} - 67}{34} \right) \times 20 = \ 70.8\]

The mode is the class with the highest frequency, \(100 - 81\) with frequency of \(67\)