9.9 The mode
Definition 9.39. The mode is the value that occurs most often. A set with one mode is unimodal; one with two is bimodal.
For the set \(4,\ 7,\ 4,\ 9,\ 4,\ 7,\ 2\) the value \(4\) appears three times, more than any other, so the mode is \(4\). In \(3,\ 5,\ 3,\ 8,\ 5,\ 9\) both \(3\) and \(5\) appear twice, so the set is bimodal with modes \(3\) and \(5\).
9.9.1 Mode for grouped data
Definition 9.40. For grouped data, \[\text {mode}=L_1+\left (\frac {\Delta _1}{\Delta _1+\Delta _2}\right )c,\] where \(L_1\) is the lower boundary of the modal class, \(c\) is its width, \(\Delta _1\) is the excess of the modal frequency over that of the class below, and \(\Delta _2\) its excess over the class above. Equivalently, \[\text {mode}=L_1+\left (\frac {f_{\text {mode}}-f_0}{2f_{\text {mode}}-f_0-f_2}\right )c.\]
Note 9.41. The two forms are the same statement. Writing \(\Delta _1=f_{\text {mode}}-f_0\) and \(\Delta _2=f_{\text {mode}}-f_2\) and adding gives \(2f_{\text {mode}}-f_0-f_2\), which is the second denominator.
Solution. The largest frequency is \(20\), in the class \(41\)–\(60\), so that is the modal class. The classes here all have the same width, so the largest frequency does identify it. \[L_1=40.5,\qquad c=20,\qquad f_0=11,\qquad f_{\text {mode}}=20,\qquad f_2=15.\] \[\Delta _1=20-11=9,\qquad \Delta _2=20-15=5.\] \[\text {mode}=40.5+\left (\frac {9}{9+5}\right )20=40.5+\frac {9}{14}\times 20 =40.5+12.86=53.36.\]
\[\therefore \quad \text {modal daily earning}\approx \text {K}53.36.\]
Note 9.43. The mode is pulled towards whichever neighbouring class is busier. Here \(\Delta _1> \Delta _2\), meaning the drop is steeper on the left, so the mode sits past the middle of the class. Had the two neighbours been equal, the formula would return the exact midpoint.
Note 9.44. For these traders the mean is K53.17, the median K53.50 and the mode K53.36 — all within a few ngwee of each other. That closeness is a sign of a roughly symmetric distribution. Where the three separate widely, the data is skewed, and which average to quote then becomes a real decision rather than a formality.
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