Find the mean (X), when ∑fi xi = 100 and ∑fi = 20

Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. It is a method of collecting and summarising the data. This has many applications from a small scale to a large scale. Whether it is the study of the population of the country or its economy, stats are used for all such data analysis.

What is Mean?

The mean (or average) of observations, is the sum of the values of all the observations divided by the total number of observations.

Mean(X) = Sum of observations/Total no. of observations

The above Formula is used to Find the mean of Raw data or Ungrouped data i.e. in which the data is given in individual points. For example, the scores of a student in the last 5 exams are given as 88, 82, 93, 85, 91. To calculate the mean, we’ll follow the following steps:

Step 1: Calculate the Sum of all observations/Entities given, i.e.

88 + 82 + 93 + 85 + 91 = 439

Step 2: Now, Divide the sum of observations by the total no. of observations,

439/5 = 87.8

Hence, the mean(X) of the above data is 87.8.

To find the Mean of Grouped data i.e. data combined together in the form of different class intervals, we use a slightly different procedure. 

If x1, x2,…, xn are observations with respective frequencies f1, f2, …, fn, then this means observation x1 occurs f1 times, x2 occurs f2 times, and so on.

Now, the sum of the values of all the observations = f1 x1 + f2 x2 + . . . + fnxn, and sum of the number of observations = f1 + f2 + . . . + fn.

So, the Mean(X) of the data is given by

Calculate mean (X), when ∑f_ix_i = 100 and ∑f_i = 20.

Solution:

Given that, ∑f_i x_i =100, ∑f_i = 20.

Mean(X) of Grouped Data = sum of the values of all the observations/ total number of observations

= ∑f_i x_i/ ∑f_i

X = 100/20 = 5

Sample Problems

Question 1: Calculate mean (X), when ∑f_i x_i =170 and ∑f_i = 50.

Solution:

Given that, ∑f_i x_i = 100,  ∑f_i = 20.

Mean(X) of Grouped Data = Sum of the values of all the observations/ total number of observations

= ∑fixi/ ∑fi

 X = 170/50 =3.04

Question 2: If ∑fixi = 45 and ∑fi = 15, then find the mean (x).

Solution: 

Given that, ∑fixi =45,  ∑f i=15.

Mean(X) of Grouped Data = Sum of the values of all the observations/ total number of observations

= ∑fixi/ ∑fi

X = 45/15 = 3

Question 3: The marks obtained by 30 students of a Class in Physics paper, out of 100 marks are given below. Find the mean(X) of the marks obtained by the students.

Marks obtained (xi) 10 20 36 40 50 56 60 70 72 80 88 92 95

Number of students (fi) 1 1 3 4 3 2 4 4 1 1 2 3 1

Solution: 

∑fixi = 10 + 20 + 108 + 160 + 150 + 112 + 240 + 280 + 72 + 80 + 176 + 276 + 95 = 1779 ,

 ∑fi = 30 (Already Given)

Mean(X) of Grouped Data = Sum of the values of all the observations/ total number of observations

Hence, the Mean of the following data is: X = 1779/30 = 59.3

Question 4: Find the mean of the following distribution:

xi = 10 30 50 70 89

fi = 7 8 10 15 10

fixi = 70 240 500 1050 890

Solution:

∑fixi = 70 + 240 + 500 + 1050 + 890 = 2750

∑fi = 7 + 8 + 10 + 15 + 10 = 55

Mean(X) of Grouped Data = Sum of the values of all the observations/ total number of observations

Therefore, x= ∑xifi / ∑fi = 2750/50 = 55

Question 5: Find the mean:

Class(xi): 0-10 10-20 20-30 30-40 40-50

Frequency(fi): 3 5 9 5 3

Solution:

fixi: 15 75 225 175 135

∑fi= 3 + 5 + 9 + 5 + 3 = 25

∑fixi= 15 + 75 + 225 + 175 + 135 = 625

Mean(X) of Grouped Data = Sum of the values of all the observations/ total number of observations

Therefore, x= ∑xifi / ∑fi = 625/25 = 25