Question

Emi computes the mean and variance for the population data set 87, 46, 90, 78, and 89. She finds the mean is 78. Her steps for finding the variance are shown below. What is the first error she made in computing the variance? Emi failed to find the difference of 89 - 78 correctly. Emi divided by N - 1 instead of N. Emi evaluated (46 - 78)2 as -(32)2. Emi forgot to take the square root of -135.6.

Answer 1

The Data set is , 87,46, 90, 78, 89.

Number of Observation = 5

Mean=
`\frac{\text{Sum of all the observation}}{\text{Total number of observation}}` =
`(87 + 46+90+78+89)/(5)=(390)/(5)= 78`

To find the Variance :

1. We have calculated mean.

2nd Step : From each Data set subtract mean in this way:

87 - 78 = 9, 46 -78 = -32, 90 - 78 = 12, 78 - 78 = 0, 89 - 78 = 11

3. Square the difference of Data and Mean.

(9)² = 81, (-32)²= 1024, (12)² = 144, 0² =0, (11)² = 121

4. Sum of all the Squared Data

81 + 1024 + 144 +0 + 121= 1370

5. Divide Sum of squared observation by 5.

We get =
`(1370)/(5) =274`

6. Variance = √274 = 16.55 (approx)

Now the value of 89 - 78 is not given .So we can't say she had made the error or not.

As we have to divide the sum of squared observations by N , not N-1.This is the first error.

Answer 2

Mean = Sum of all the observations/ Number of observations = (87+46+90+78+89)/5 = 78

Variance = SD^2 ------ SD = Standard deviation

It means that with Variance, square root is never taken.

Therefore,
Variance = Summation of square of differences between observation and the mean divided by the number of observations.

That is,
Variance = {(87-78)^2 + (46-78)^2 + (90-78)^2 + (78-78)^2 + (89-78)^2}/5 = 274