A STEP BY STEP EASY METHOD OF CALCULATING MODE IS ILLUSTRATED HERE. BOTH DISCRETE AND GROUPED FREQUENCY DISTRIBUTION IS EXPLAINED IN AN EASY MANNER. THIS IS A QUICK METHOD OF LEARNING HOW TO CALCULATE MODE.

1. GRADE 11 MATH
COLLEGE MATH SUMAN MATHEWS

2. HERE, WE LEARN HOW TO CALCULATE MODE FOR A DISCRETE SET OF OBSERVATIONS
AND ALSO FOR A FREQUENCY DISTRIBUTION.
THE FORMULAE FOR COMBINED MEAN AND COMBINED STANDARD DEVIATION ARE
ALSO DISCUSSED.

3. FOR A DISCRETE SET OF VALUES OR AN UNGROUPED FREQUENCY DISTRIBUTION, MODE IS
THE VALUE WITH THE LARGEST FREQUENCY.
FOR A GROUPED FREQUENCY DISTRIBUTION,
𝑀𝑂𝐷𝐸 = 𝑙 + (
𝑓𝑚 − 𝑓 𝑚−1
2𝑓𝑚 − 𝑓 𝑚−1 − 𝑓 𝑚+1
) × 𝑐
l = lower limit of the modal class
𝑓𝑚 = 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑜𝑑𝑎𝑙 𝑐𝑙𝑎𝑠𝑠
𝑓 𝑚−1 = 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑙𝑎𝑠𝑠 𝑝𝑟𝑒𝑐𝑒𝑒𝑑𝑖𝑛𝑔 𝑡ℎ𝑒 𝑚𝑜𝑑𝑎𝑙 𝑐𝑙𝑎𝑠𝑠
𝑓 𝑚+1 = 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑙𝑎𝑠𝑠 𝑠𝑢𝑐𝑐𝑒𝑒𝑑𝑖𝑛𝑔 𝑡ℎ𝑒 𝑚𝑜𝑑𝑎𝑙 𝑐𝑙𝑎𝑠𝑠
c = width of the modal class

4. Mode = 3 median – 2 mean
Question 1
Find the mode of
2,1, 1, 2, 3, 2, 5, 4, 6, 4, 1, 2, 3
Mode = 2

5. Question 2
find the mode of the following frequency distribution
x f
14 14
15 26
16 18
17 9
18 2
19 1
20 1
Mode = 15( value with the highest frequency)

6. Question 3
The daily wages of 30 employees are as follows
C I f
0 -10 1
10 -20 8
20- 30 10
30- 40 5
40- 50 4
50- 60 2
𝑚𝑜𝑑𝑒 = 20 +
10 − 8
20 − 8 − 5
× 10 = 22.85
𝑀𝑂𝐷𝐸 = 𝑙 + (
𝑓𝑚 − 𝑓 𝑚−1
2𝑓𝑚 − 𝑓 𝑚−1 − 𝑓 𝑚+1
) × 𝑐

7. QUESTION 4
FIND a, b FROM THE FOLLOWING DATA
C I f
10 - 20 5
20 -30 a
30 -40 15
40 -50 b
50 -60 7
Mode = 37
Total frequency = 47
27 + a + b = 47
a + b = 20
b = 20 -a

8. 37 = 30 +
15 − 𝑎
30 − 𝑎 − 𝑏
× 10
37 = 30 +
15 − 𝑎
30 − 𝑎 − 20 − 𝑎
10
37 = 30 +
15 − 𝑎
10
10
37 = 45 - a
𝑎 = 8, 𝑏 = 12

9. Combined mean and
standard deviation
𝐼𝑓 𝑎 𝑔𝑟𝑜𝑢𝑝 𝑜𝑓 𝑛1 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 ℎ𝑎𝑠 𝑚𝑒𝑎𝑛 𝑥1 𝑎𝑛𝑑 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝜎1 𝑎𝑛𝑑
𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑔𝑟𝑜𝑢𝑝 𝑜𝑓 𝑛2 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 ℎ𝑎𝑠 𝑚𝑒𝑎𝑛 𝑥2 𝑎𝑛𝑑 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝜎2, 𝑡ℎ𝑒𝑛
𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 𝑚𝑒𝑎𝑛 𝑥 =
𝑛1 𝑥1 + 𝑛2 𝑥2
𝑛1 + 𝑛2
𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝜎2
=
𝑛1 𝜎1
2
+ 𝑛2 𝜎2
2
+ 𝑛1 𝑥1 − 𝑥 2
+ 𝑛2 𝑥2 − 𝑥 2
𝑛1 + 𝑛2

10. A group of 35 values has mean 80 and standard deviation 4. A second sample of 65 values has mean 70
And standard deviation 3. Find the mean and standard deviation of the combined sample of 100 values
𝑛1 = 35, 𝑥1 = 80, 𝜎1 = 4
𝑛2 = 65, 𝑥2 = 70, 𝜎2 = 3
𝑥 =
35 × 80 + (65 × 70)
35 + 65
= 73.5

11. 𝜎2
=
35 16 + 65 9 + 35 80 − 73.5 2
+ 65 70 − 73.5 2
100
𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝜎2
=
𝑛1 𝜎1
2
+ 𝑛2 𝜎2
2
+ 𝑛1 𝑥1 − 𝑥 2
+ 𝑛2 𝑥2 − 𝑥 2
𝑛1 + 𝑛2
𝜎2
= 34.2
𝜎 = 5.848

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