Statistics Examples

\begin{matrix} Class & Frequency 12 - 17 & 3 18 - 23 & 6 24 - 29 & 4 30 - 35 & 2 \end{matrix}

$\begin{array}{c} Class & Frequency \\ 12 - 17 & 3 \\ 18 - 23 & 6 \\ 24 - 29 & 4 \\ 30 - 35 & 2 \end{array}$

Step 1

Find the midpoint

M

$M$ for each group.

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Step 1.1

The lower limit for every class is the smallest value in that class. On the other hand, the upper limit for every class is the greatest value in that class.

\begin{matrix} Class & Frequency (f) & Lower Limits & Upper Limits 12 - 17 & 3 & 12 & 17 18 - 23 & 6 & 18 & 23 24 - 29 & 4 & 24 & 29 30 - 35 & 2 & 30 & 35 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Lower Limits & Upper Limits \\ 12 - 17 & 3 & 12 & 17 \\ 18 - 23 & 6 & 18 & 23 \\ 24 - 29 & 4 & 24 & 29 \\ 30 - 35 & 2 & 30 & 35 \end{array}$

Step 1.2

The class midpoint is the lower class limit plus the upper class limit divided by

2

$2$ .

\begin{matrix} Class & Frequency (f) & Lower Limits & Upper Limits & Midpoint (M) 12 - 17 & 3 & 12 & 17 & \frac{12 + 17}{2} 18 - 23 & 6 & 18 & 23 & \frac{18 + 23}{2} 24 - 29 & 4 & 24 & 29 & \frac{24 + 29}{2} 30 - 35 & 2 & 30 & 35 & \frac{30 + 35}{2} \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Lower Limits & Upper Limits & Midpoint (M) \\ 12 - 17 & 3 & 12 & 17 & \frac{12 + 17}{2} \\ 18 - 23 & 6 & 18 & 23 & \frac{18 + 23}{2} \\ 24 - 29 & 4 & 24 & 29 & \frac{24 + 29}{2} \\ 30 - 35 & 2 & 30 & 35 & \frac{30 + 35}{2} \end{array}$

Step 1.3

Simplify all the midpoint column.

\begin{matrix} Class & Frequency (f) & Lower Limits & Upper Limits & Midpoint (M) 12 - 17 & 3 & 12 & 17 & 14.5 18 - 23 & 6 & 18 & 23 & 20.5 24 - 29 & 4 & 24 & 29 & 26.5 30 - 35 & 2 & 30 & 35 & 32.5 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Lower Limits & Upper Limits & Midpoint (M) \\ 12 - 17 & 3 & 12 & 17 & 14.5 \\ 18 - 23 & 6 & 18 & 23 & 20.5 \\ 24 - 29 & 4 & 24 & 29 & 26.5 \\ 30 - 35 & 2 & 30 & 35 & 32.5 \end{array}$

Step 1.4

Add the midpoints column to the original table.

\begin{matrix} Class & Frequency (f) & Midpoint (M) 12 - 17 & 3 & 14.5 18 - 23 & 6 & 20.5 24 - 29 & 4 & 26.5 30 - 35 & 2 & 32.5 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) \\ 12 - 17 & 3 & 14.5 \\ 18 - 23 & 6 & 20.5 \\ 24 - 29 & 4 & 26.5 \\ 30 - 35 & 2 & 32.5 \end{array}$

\begin{matrix} Class & Frequency (f) & Midpoint (M) 12 - 17 & 3 & 14.5 18 - 23 & 6 & 20.5 24 - 29 & 4 & 26.5 30 - 35 & 2 & 32.5 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) \\ 12 - 17 & 3 & 14.5 \\ 18 - 23 & 6 & 20.5 \\ 24 - 29 & 4 & 26.5 \\ 30 - 35 & 2 & 32.5 \end{array}$

Step 2

Calculate the square of each group midpoint

M^{2}

$M^{2}$ .

\begin{matrix} Class & Frequency (f) & Midpoint (M) & M^{2} 12 - 17 & 3 & 14.5 & {14.5}^{2} 18 - 23 & 6 & 20.5 & {20.5}^{2} 24 - 29 & 4 & 26.5 & {26.5}^{2} 30 - 35 & 2 & 32.5 & {32.5}^{2} \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} \\ 12 - 17 & 3 & 14.5 & {14.5}^{2} \\ 18 - 23 & 6 & 20.5 & {20.5}^{2} \\ 24 - 29 & 4 & 26.5 & {26.5}^{2} \\ 30 - 35 & 2 & 32.5 & {32.5}^{2} \end{array}$

Step 3

Simplify the

M^{2}

$M^{2}$ column.

\begin{matrix} Class & Frequency (f) & Midpoint (M) & M^{2} 12 - 17 & 3 & 14.5 & 210.25 18 - 23 & 6 & 20.5 & 420.25 24 - 29 & 4 & 26.5 & 702.25 30 - 35 & 2 & 32.5 & 1056.25 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} \\ 12 - 17 & 3 & 14.5 & 210.25 \\ 18 - 23 & 6 & 20.5 & 420.25 \\ 24 - 29 & 4 & 26.5 & 702.25 \\ 30 - 35 & 2 & 32.5 & 1056.25 \end{array}$

Step 4

Multiply each midpoint squared by its frequency

f

$f$ .

\begin{matrix} Class & Frequency (f) & Midpoint (M) & M^{2} & f \cdot M^{2} 12 - 17 & 3 & 14.5 & 210.25 & 3 \cdot 210.25 18 - 23 & 6 & 20.5 & 420.25 & 6 \cdot 420.25 24 - 29 & 4 & 26.5 & 702.25 & 4 \cdot 702.25 30 - 35 & 2 & 32.5 & 1056.25 & 2 \cdot 1056.25 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} & f \cdot M^{2} \\ 12 - 17 & 3 & 14.5 & 210.25 & 3 \cdot 210.25 \\ 18 - 23 & 6 & 20.5 & 420.25 & 6 \cdot 420.25 \\ 24 - 29 & 4 & 26.5 & 702.25 & 4 \cdot 702.25 \\ 30 - 35 & 2 & 32.5 & 1056.25 & 2 \cdot 1056.25 \end{array}$

Step 5

Simplify the

f \cdot M^{2}

$f \cdot M^{2}$ column.

\begin{matrix} Class & Frequency (f) & Midpoint (M) & M^{2} & f \cdot M^{2} 12 - 17 & 3 & 14.5 & 210.25 & 630.75 18 - 23 & 6 & 20.5 & 420.25 & 2521.5 24 - 29 & 4 & 26.5 & 702.25 & 2809 30 - 35 & 2 & 32.5 & 1056.25 & 2112.5 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} & f \cdot M^{2} \\ 12 - 17 & 3 & 14.5 & 210.25 & 630.75 \\ 18 - 23 & 6 & 20.5 & 420.25 & 2521.5 \\ 24 - 29 & 4 & 26.5 & 702.25 & 2809 \\ 30 - 35 & 2 & 32.5 & 1056.25 & 2112.5 \end{array}$

Step 6

Find the sum of all frequencies. In this case, the sum of all frequencies is

n = 3, 6, 4, 2 = 15

$n = 3, 6, 4, 2 = 15$ .

\sum f = n = 15

$\sum_{}^{} f = n = 15$

Step 7

Find the sum of

f \cdot M^{2}

$f \cdot M^{2}$ column. In this case,

630.75 + 2521.5 + 2809 + 2112.5 = 8073.75

$630.75 + 2521.5 + 2809 + 2112.5 = 8073.75$ .

\sum f \cdot M^{2} = 8073.75

$\sum_{}^{} f \cdot M^{2} = 8073.75$

Step 8

Find the mean

μ

$μ$ .

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Step 8.1

Find the midpoint

M

$M$ for each class.

\begin{matrix} Class & Frequency (f) & Midpoint (M) 12 - 17 & 3 & 14.5 18 - 23 & 6 & 20.5 24 - 29 & 4 & 26.5 30 - 35 & 2 & 32.5 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) \\ 12 - 17 & 3 & 14.5 \\ 18 - 23 & 6 & 20.5 \\ 24 - 29 & 4 & 26.5 \\ 30 - 35 & 2 & 32.5 \end{array}$

Step 8.2

Multiply the frequency of each class by the class midpoint.

\begin{matrix} Class & Frequency (f) & Midpoint (M) & f \cdot M 12 - 17 & 3 & 14.5 & 3 \cdot 14.5 18 - 23 & 6 & 20.5 & 6 \cdot 20.5 24 - 29 & 4 & 26.5 & 4 \cdot 26.5 30 - 35 & 2 & 32.5 & 2 \cdot 32.5 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & f \cdot M \\ 12 - 17 & 3 & 14.5 & 3 \cdot 14.5 \\ 18 - 23 & 6 & 20.5 & 6 \cdot 20.5 \\ 24 - 29 & 4 & 26.5 & 4 \cdot 26.5 \\ 30 - 35 & 2 & 32.5 & 2 \cdot 32.5 \end{array}$

Step 8.3

Simplify the

f \cdot M

$f \cdot M$ column.

\begin{matrix} Class & Frequency (f) & Midpoint (M) & f \cdot M 12 - 17 & 3 & 14.5 & 43.5 18 - 23 & 6 & 20.5 & 123 24 - 29 & 4 & 26.5 & 106 30 - 35 & 2 & 32.5 & 65 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & f \cdot M \\ 12 - 17 & 3 & 14.5 & 43.5 \\ 18 - 23 & 6 & 20.5 & 123 \\ 24 - 29 & 4 & 26.5 & 106 \\ 30 - 35 & 2 & 32.5 & 65 \end{array}$

Step 8.4

Add the values in the

f \cdot M

$f \cdot M$ column.

43.5 + 123 + 106 + 65 = 337.5

$43.5 + 123 + 106 + 65 = 337.5$

Step 8.5

Add the values in the frequency column.

n = 3 + 6 + 4 + 2 = 15

$n = 3 + 6 + 4 + 2 = 15$

Step 8.6

The mean (mu) is the sum of

f \cdot M

$f \cdot M$ divided by

n

$n$ , which is the sum of frequencies.

μ = \frac{\sum f \cdot M}{\sum f}

$μ = \frac{\sum_{}^{} f \cdot M}{\sum_{}^{} f}$

Step 8.7

The mean is the sum of the product of the midpoints and frequencies divided by the total of frequencies.

μ = \frac{337.5}{15}

$μ = \frac{337.5}{15}$

Step 8.8

Simplify the right side of

μ = \frac{337.5}{15}

$μ = \frac{337.5}{15}$ .

22.5

$22.5$

22.5

$22.5$

Step 9

The equation for the standard deviation is

S^{2} = \frac{\sum f \cdot M^{2} - n {(μ)}^{2}}{n - 1}

$S^{2} = \frac{\sum_{}^{} f \cdot M^{2} - n {(μ)}^{2}}{n - 1}$ .

S^{2} = \frac{\sum f \cdot M^{2} - n {(μ)}^{2}}{n - 1}

$S^{2} = \frac{\sum_{}^{} f \cdot M^{2} - n {(μ)}^{2}}{n - 1}$

Step 10

Substitute the calculated values into

S^{2} = \frac{\sum f \cdot M^{2} - n {(μ)}^{2}}{n - 1}

$S^{2} = \frac{\sum_{}^{} f \cdot M^{2} - n {(μ)}^{2}}{n - 1}$ .

S^{2} = \frac{8073.75 - 15 {(22.5)}^{2}}{15 - 1}

$S^{2} = \frac{8073.75 - 15 {(22.5)}^{2}}{15 - 1}$

Step 11

Simplify the right side of

S^{2} = \frac{8073.75 - 15 {(22.5)}^{2}}{15 - 1}

$S^{2} = \frac{8073.75 - 15 {(22.5)}^{2}}{15 - 1}$ to get the variance

S^{2} = 34. ¯ ¯¯¯¯¯¯¯¯¯¯ ¯ 285714

$S^{2} = 34.\overline{285714}$ .

34.28571428

$34.28571428$

Step 12

The standard deviation is the square root of the variance

34. ¯ ¯¯¯¯¯¯¯¯¯¯ ¯ 285714

$34.\overline{285714}$ . In this case, the standard deviation is

5.85540043

$5.85540043$ .

5.85540043

$5.85540043$

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