Question

The data in this question will be utilized for the next three questions. The number of pets a group of students have are given below.Solute deviation?

Answer 1

Question

The data in this question will be utilized for the next three questions. The number of pets a group of students has, are given below.

Number of pets:        0-2    2-4   4-6   6-8   8-10   10-12    12-14

Number of students:  1        2      1        5       6         2           3

Calculate

(1) Mean      (2) Sample Variance    (3) Sample Standard Deviation

Answer:

$(1)$ $\bar x = 8.1$

$(2)$ $\sigma^2 = 6.341$

$(3)$ $\sigma = 2.518$

Step-by-step explanation:

Given

The above data

First, we calculate the class midpoint (x)

Pets:  $0-2$    $2-4$   $4-6$   $6-8$   $8-10$   $10-12$    $12-14$

x           1           3          5       7           9          11          13

f:           1            2         1        5           6         2           3

The class midpoint (x) is calculated by the average of each group.

For pets: 0 - 2.

$x = \frac{0 + 2}{2}= \frac{2}{2} = 1$

The same is done for other groups.

Solving (a): Mean

$\bar x = \frac{\sum fx}{\sum f}$

This gives:

$\bar x = \frac{1 * 1 + 3 * 2 + 5 * 1 + 7 * 5 + 9 * 6 + 11 * 2 + 13 * 3}{1 + 2 + 1 + 5 + 6 + 2 + 3}$

$\bar x = \frac{162}{20}$

$\bar x = 8.1$

Solving (b): Sample Variance

This is calculated as:

$\sigma^2 = \frac{\sum(x_i - \bar x)^2}{\sum f - 1}$

So:

$\sigma^2 = \frac{(1 - 8.1)^2+(3 - 8.1)^2+(5 - 8.1)^2+(7 - 8.1)^2+(9 - 8.1)^2+(11 - 8.1)^2+(13 - 8.1)^2}{20 - 1}$

$\sigma^2 = \frac{120.47}{19}$

$\sigma^2 = 6.341$

Solving (c): The sample standard deviation

This is calculated as:

$\sigma = \sqrt{\sigma^2$

$\sigma = \sqrt{6.341$

$\sigma = 2.518$