Heights of students of Class X are given in the following frequency distribution:
Height (in cm) | 150 - 155 | 155 - 160 | 160 - 165 | 165 - 170 | 170 - 175 |
Number of students | 15 | 8 | 20 | 12 | 5 |
Find the modal height.
Also, find the mean height. Compare and interpret the two measures of central tendency.
Here, the maximum class frequency is 20.
The class corresponding to this frequency is the modal class. ⇒ modal class = 160 - 165
∴ lower limit of the modal class (l) = 160
Modal class size (h) = 5
Frequency of the modal class (f1) = 20
Frequency of class preceding the modal class (f0) = 8
Frequency of class succeeding the modal (f2) = 12
Mode is given by,
⇒ 
⇒ 
⇒ Mode = 160 + 3 = 163
Hence, the mode is 163 cm.
Mode represents frequency, hence 163 cm is the height of maximum number of students.
To find the mean, we will solve by using direct method.
HEIGHT (cm) | MID - POINT(xi) | NUMBER OF STUDENTS(fi) | fixi |
150 - 155 | 152.5 | 15 | 2287.5 |
155 - 160 | 157.5 | 8 | 1260 |
160 - 165 | 162.5 | 20 | 3250 |
165 - 170 | 167.5 | 12 | 2010 |
170 - 175 | 172.5 | 5 | 862.5 |
TOTAL | 60 | 9670 |
We have got
Σfi = 60 & Σfixi = 9670
∵ mean is given by
⇒ 
⇒ 
Thus, mean is 161.17
Mean represents average, thus 161.7 cm is the average height of all the students.