Mode of Grouped Data

Mode of Grouped Data — the Modal Class

The mode is the value that occurs most often. For ungrouped data you just spot the value with the highest frequency. For grouped data you cannot see individual values, so first find the modal class — the class with the highest frequency. The mode is a value inside that class.

The Mode Formula

$$\boxed{\text{Mode} = l + \left(\dfrac{f_1 - f_0}{2f_1 - f_0 - f_2}\right)\times h}$$

where

• $l$ = lower limit of the modal class,
• $f_1$ = frequency of the modal class,
• $f_0$ = frequency of the class just before the modal class,
• $f_2$ = frequency of the class just after the modal class,
• $h$ = class size.

Key Point: Identify the modal class first (highest frequency), then read off $l, f_1, f_0, f_2, h$ and substitute carefully — the denominator is $2f_1 - f_0 - f_2$.

Reading Off the Values

The commonest slips are in $f_0$ and $f_2$. Remember:

• $f_0$ is the frequency of the class immediately above the modal class in the table (the preceding class),
• $f_2$ is the frequency of the class immediately below it (the succeeding class).

If the modal class is the first class, take $f_0 = 0$; if it is the last, take $f_2 = 0$.

Key Point: Line up the three frequencies $f_0, f_1, f_2$ in order and the formula almost fills itself in.

Mode vs Mean

The mode tells you the most typical value; the mean tells you the average. They answer different questions. For the marks data, the mode is 52 (most students scored around there) while the mean is 62 (the overall average) — both are useful, depending on what you want to know.

[Board Important] "Most frequent / most common / typical" $\Rightarrow$ mode. "Average / per student" $\Rightarrow$ mean.

Solved Examples

Example 1: Mode of the marks data

Find the mode: classes $10\text{-}25,25\text{-}40,40\text{-}55,55\text{-}70,70\text{-}85,85\text{-}100$ with frequencies $2,3,7,6,6,6$.

Solution:

1. Highest frequency is 7 $\Rightarrow$ modal class $40\text{-}55$. So $l=40$, $f_1=7$, $f_0=3$, $f_2=6$, $h=15$.
2. Mode $= 40 + \dfrac{7-3}{2(7)-3-6}\times 15 = 40 + \dfrac{4}{5}\times 15 = 40 + 12 = 52$.

Final Answer: Mode $= 52$.

Takeaway: Denominator $=2f_1-f_0-f_2=14-9=5$.

Example 2: Family size

Find the mode: family size $1\text{-}3,3\text{-}5,5\text{-}7,7\text{-}9,9\text{-}11$ with frequencies $7,8,2,2,1$.

Solution:

1. Highest frequency is 8 $\Rightarrow$ modal class $3\text{-}5$; $l=3$, $f_1=8$, $f_0=7$, $f_2=2$, $h=2$.
2. Mode $= 3 + \dfrac{8-7}{2(8)-7-2}\times 2 = 3 + \dfrac{1}{7}\times 2 = 3 + 0.286 = 3.286$.

Final Answer: Mode $\approx 3.29$.

Takeaway: Even when $f_0$ is close to $f_1$, follow the formula exactly.

Example 3: First class is modal

Find the mode: $0\text{-}10,10\text{-}20,20\text{-}30$ with frequencies $15, 6, 4$.

Solution:

1. Modal class is $0\text{-}10$ (frequency 15). $l=0$, $f_1=15$, $f_0=0$ (no class before), $f_2=6$, $h=10$.
2. Mode $= 0 + \dfrac{15-0}{2(15)-0-6}\times 10 = \dfrac{15}{24}\times 10 = 6.25$.

Final Answer: Mode $= 6.25$.

Takeaway: When the modal class is first, use $f_0 = 0$.

Example 4: Identify only the modal class

For the distribution $30\text{-}40,40\text{-}50,50\text{-}60,60\text{-}70$ with frequencies $5, 12, 20, 9$, name the modal class and $f_0, f_1, f_2$.

Solution:

1. Highest frequency 20 $\Rightarrow$ modal class $50\text{-}60$.
2. $f_1 = 20$, $f_0 = 12$ (class $40\text{-}50$), $f_2 = 9$ (class $60\text{-}70$).

Final Answer: Modal class $50\text{-}60$; $f_0=12, f_1=20, f_2=9$.

Takeaway: Getting $f_0$ and $f_2$ right is half the battle.