Question

The mean of 40 observations is 160. While cross checking it was
found that the observation 165 has wrongly been read as 125. Find the new correct mean.

Answer 1

Given :-

The mean of 40 observations is 160.

While cross checking it was found that the observation 165 has wrongly been read as 125.

To find :-

The new correct mean.

Solution :-

Given that

Number of observations = 40

The Mean of 40 observations = 160

We know that

Mean = Sum of observations / Number of observations

=> 160 = Sum of observations / 40

=> Sum of observations = 160 × 40

Sum of 40 observations = 6400

The correct observation = 165

It is read as 125

Therefore, The reduced in the total

= 165-125 = 40

40 should be added to the sum of all observations for rectifying the mistake

Therefore, Sum of observations

= 6400+40

= 6440

Now

The New correct Mean

= 6440/40

= 161

The New correct Mean = 161

Alternative Method :-

" The mean of 'n' quantities is A but while taking the values,one quantity P is erroreously read as Q then the new correct mean = A-[(Q-P)/n].

We have,

A = 160

Q = 125

P = 165

n = 40

Therefore, New Correct Mean

= 160- [ (125-165)/40]

= 160-(-40/40)

= 160-(-1)

= 160+1

= 161

The New correct Mean = 161

Answer :-

The new correct mean of the observations is 161

Used formulae:-

♦ Mean = Sum of observations / Number of observations

♦ The mean of 'n' quantities is A but while taking the values,one quantity P is erroreously read as Q then the new correct mean = A-[(Q-P)/n].

Answer 2

INFORMATION PROVIDED :-

• mean of 40 observations is 160
• the observation 165 has wrongly been read as 125

TO FIND :-

• Find the new correct mean = ?

CONCEPT USED :-

• formula to find mean = (sum of all observations)/( total Number of observations)

UNDERSTANDING CONCEPT :-

• We solve this question by first considering the formula for mean, mean = (sum of all observations)/( total Number of observations)Then we substitute the given value of mean and number of observations and solve it to find the value of the sum of observations. Then we subtract the observation 125 and add 165 to the sum and find the value of the corrected sum of observations. Then we use the formula for mean again to find the correct mean.

SOLUTION :-

We have,

n = Number of observations 40, Mean = 160

Mean = Sum of the observations/Number of observations

160 = Sum of the observations/40

160 × 40 = Sum of the observations.

• Thus, incorrect sum of observations = 160 × 40

Now,

Correct sum of the observations = Incorrect sum of the observations - Incorrect observation + correct observation

Correct sum of the observations

= 160 × 40 - 125 + 165

Correct sum of the observations

= 6400 + 40 = 6440

Correct mean

= Corect sum of the observations/Number of observations

= 6400 /40 = 161

new correct mean = 161

ADDITIONAL INFORMATION :-

• The common mistake one makes while solving this problem is one might take the total number of observations at the end as 41 thinking that we are adding 165 to the before sum. But it is wrong because we are adding 165 but subtracting 125. So, the number of observations remains 40.