The Value That Occurs Most Often
A shoe shop owner gains very little from knowing the average shoe size of a town. What she really needs is the size that most customers ask for, because that is the size she must stock in the largest number. In statistics, this most popular value has a name. It is called the mode.
Think of the temperatures of a city recorded over 20 days, the marks of a class, or the sizes of shoes sold in one day. In each of these, some values turn up again and again. The value that turns up the greatest number of times is the mode. It is written with the symbol \( M_o \).
Mode: The mode of a set of data is the value that is repeated the maximum number of times. In other words, it is the value with the highest frequency.
Like the mean and the median, the mode is a measure of central tendency. For ungrouped data, however, finding it needs no adding, no dividing and no search for a middle value. You only have to count how many times each value appears.
Mode of an Individual Series
In an individual series, every observation is written out one by one. To find the mode, count how often each value appears and choose the value with the largest count. Arranging the data in ascending order first helps, because equal values then sit side by side and are easy to count.
Here are the shoe sizes of ten students: 5, 6, 7, 6, 8, 6, 5, 7, 6, 9. Counting how many times each size appears gives this table.
| Shoe size | Number of students |
| 5 | 2 |
| 6 | 4 |
| 7 | 2 |
| 8 | 1 |
| 9 | 1 |
Size 6 appears 4 times, more than any other size. So the mode of this data is 6. A shop selling shoes to these students should keep more pairs of size 6 than of any other size.
A set of data can have more than one mode. In 2, 3, 3, 5, 7, 7, 9 both 3 and 7 appear twice, so the data has two modes, 3 and 7. If every value appears the same number of times, as in 4, 8, 11, 15, no value is repeated more than the others and the data has no mode.
Mode of a Discrete Series
In a discrete series the counting has already been done. Each value is written once, with its frequency beside it telling how many times it occurred. So look for the highest frequency. The value that owns that frequency is the mode.
The number of members in each of 50 families of a village is shown below.
| Family size | Number of families |
| 2 | 5 |
| 3 | 12 |
| 4 | 20 |
| 5 | 9 |
| 6 | 4 |
The highest frequency is 20, and it belongs to a family size of 4. So the mode is 4 members. In this village, more families have four members than any other number.
The mode is the value, not its frequency. In the table above the answer is 4 members, never 20. The frequency only tells you which value to pick.
Mode of a Continuous Series
In a continuous series the data is grouped into class intervals such as 30-40, 40-50 and 50-60, and only the frequency of each class is given. The individual values can no longer be seen. We can tell which class holds the most values, but not which exact value inside that class was repeated most.
Modal class: In a continuous series, the class interval with the highest frequency is called the modal class. The mode lies somewhere inside this class.
The mode is not simply the lower limit of the modal class, and it is not simply its middle either. Where it falls inside the class depends on the classes on each side. If the class just after the modal class holds many values and the class just before holds few, the mode is pulled towards the upper limit. If it is the other way round, it is pulled towards the lower limit. A formula takes all of this into account.
The Formula and Its Symbols
The mode of grouped data is found in the following steps:
- (a) Since the mode is about how often values repeat, first find the class interval with the highest frequency. This is the modal class.
- (b) Write down the frequency of the modal class as \( f_1 \), the frequency of the class preceding it as \( f_0 \), and the frequency of the class succeeding it as \( f_2 \).
- (c) Find the lower limit \( L \) of the modal class and the width \( h \) of its class interval.
- (d) Put these values into the formula below to find the mode.
\[ M_o = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h \]
Where,
- \( L \) = lower limit of the modal class
- \( f_1 \) = frequency of the modal class
- \( f_0 \) = frequency of the class preceding the modal class
- \( f_2 \) = frequency of the class succeeding the modal class
- \( h \) = width (size) of the modal class interval, that is, its upper limit minus its lower limit
\( f_0 \) belongs to the class BEFORE the modal class and \( f_2 \) to the class AFTER it. Swap them and the top of the fraction changes, so the answer comes out wrong. In the same way, \( L \) is the lower limit of the modal class, not the first number in the whole table.
Worked Example 1: Weights of Students
The weights of 40 students are given in the table below. Find the mode.
| Weight (kg) | Number of students |
| 30-40 | 3 |
| 40-50 | 5 |
| 50-60 | 7 |
| 60-70 | 11 |
| 70-80 | 10 |
| 80-90 | 3 |
| 90-100 | 1 |
Step 1: Finding the modal class. The highest frequency is 11, so the modal class is 60-70.
Step 2: Reading the values the formula needs from the modal class and its two neighbours:
\[ \begin{aligned} L &= 60 \\ f_1 &= 11 \\ f_0 &= 7 \\ f_2 &= 10 \\ h &= 70 - 60 = 10 \end{aligned} \]
Step 3: Substituting these values into the formula:
\[ M_o = 60 + \frac{11 - 7}{2 \times 11 - 7 - 10} \times 10 \]
Step 4: Simplifying the top and the bottom of the fraction. Since \( 11 - 7 = 4 \) and \( 22 - 7 - 10 = 5 \), this becomes:
\[ M_o = 60 + \frac{4}{5} \times 10 \]
Step 5: Multiplying, then adding to the lower limit:
\[ M_o = 60 + 8 = 68 \]
Hence the mode is 68 kg. The weight around which most of these students cluster is 68 kg.
Why the Mode Leans Towards the Bigger Neighbour
The middle of the modal class 60-70 is 65. The mode we found is 68, which is not in the middle but leans towards 70. The neighbours explain why. The class before has 7 students and the class after has 10. The side with more students pulls the mode towards itself. If the two neighbours had equal frequencies, that is \( f_0 = f_2 \), the formula would give exactly the middle of the modal class.
A quick way to check your answer: the mode of a continuous series always lies inside the modal class, between its lower and upper limits. If your answer falls outside the class, a value was put into the formula wrongly.
Worked Example 2: Marks in a Test
The marks obtained by 45 students in a test are given below. Find the mode.
| Marks obtained | Number of students |
| 0-10 | 4 |
| 10-20 | 9 |
| 20-30 | 15 |
| 30-40 | 11 |
| 40-50 | 6 |
Step 1: Finding the modal class. The highest frequency is 15, so the modal class is 20-30.
Step 2: Reading the values the formula needs:
\[ \begin{aligned} L &= 20 \\ f_1 &= 15 \\ f_0 &= 9 \\ f_2 &= 11 \\ h &= 30 - 20 = 10 \end{aligned} \]
Step 3: Substituting into the formula:
\[ M_o = 20 + \frac{15 - 9}{2 \times 15 - 9 - 11} \times 10 \]
Step 4: Simplifying the fraction. Since \( 15 - 9 = 6 \) and \( 30 - 9 - 11 = 10 \), this becomes:
\[ M_o = 20 + \frac{6}{10} \times 10 \]
Step 5: Multiplying, then adding to the lower limit:
\[ M_o = 20 + 6 = 26 \]
Hence the mode is 26 marks. It lies inside the modal class 20-30, a little above its middle value 25, because the class after it (11 students) holds more students than the class before it (9 students).
When the Modal Class Is at the Edge
Sometimes the highest frequency belongs to the very first class of the table. Then there is no class before it, so we take \( f_0 = 0 \). In the same way, if the modal class is the last class, there is no class after it and we take \( f_2 = 0 \). The rest of the method does not change.
For example, suppose the classes 0-10, 10-20, 20-30 and 30-40 have frequencies 12, 8, 5 and 3. The modal class is 0-10, so \( L = 0 \), \( f_1 = 12 \), \( f_0 = 0 \), \( f_2 = 8 \) and \( h = 10 \). Putting these into the formula:
\[ M_o = 0 + \frac{12 - 0}{2 \times 12 - 0 - 8} \times 10 \]
Since the bottom is \( 24 - 8 = 16 \), simplifying gives:
\[ M_o = \frac{12}{16} \times 10 = 7.5 \]
The answer 7.5 lies inside the modal class 0-10, so it passes the check.
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