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. mc008-1.jpg 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

Well let’s find the actually mean

First we add up all the numbers

87+46+90+78+89 = 390

Next we divide 390 by how many numbers there are.

390/5 = 78

Therefore the mean is 78

Emi failed to find the difference

Answer 2

Variance is calculated as the average of squared differences of the data observations from the mean. The variance of the given data set is obtained as $\sigma^2 = 274$

How to calculate variance of a given data set?

Let the data set be $x_1, x_2, ... , x_n$ (n sized)

Their  mean is $\overline{x} = \dfrac{\sum_{i=1}^n{x_i}}{n}$

Then the variance of the data set is given by

$\sigma ^2 = \sum_{i=1}^n{\dfrac{(x_i - \overline{x})^2}{{n}}$

Using the above formula, as the data set is given

87, 46, 90, 78, and 89, and mean is 78, thus,

$n = 5\\\overline{x} = 78$

Thus,

$\sigma ^2 = \sum_{i=1}^n{\dfrac{(x_i - \overline{x})^2}{{n}}}\\\\\sigma^2 = \dfrac{(87-78)^2 + (46 - 78)^2 + (90-78)^2 + (78-78)^2 + (89-78)^2 }{5}\\\\\sigma^2 = \dfrac{1370}{5} = 274$

Thus,

The variance of the given data set is obtained as $\sigma^2 = 274$

Learn more about variance here:

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