Finite Math Examples Frequency Distribution

Find the Variance of the Frequency Table

$\begin{array}{c} Class & Frequency \\ 90 - 99 & 4 \\ 80 - 89 & 6 \\ 70 - 79 & 4 \\ 60 - 69 & 3 \\ 50 - 59 & 2 \\ 40 - 49 & 1 \end{array}$

Step 1

Reorder the classes with their related frequencies in an ascending order (lowest number to highest), which is the most common.

$\begin{array}{c} Class & Frequency (f) \\ 40 - 49 & 1 \\ 50 - 59 & 2 \\ 60 - 69 & 3 \\ 70 - 79 & 4 \\ 80 - 89 & 6 \\ 90 - 99 & 4 \end{array}$

Step 2

Find the midpoint $M$ for each group.

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Step 2.1

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.

$\begin{array}{c} Class & Frequency (f) & Lower Limits & Upper Limits \\ 40 - 49 & 1 & 40 & 49 \\ 50 - 59 & 2 & 50 & 59 \\ 60 - 69 & 3 & 60 & 69 \\ 70 - 79 & 4 & 70 & 79 \\ 80 - 89 & 6 & 80 & 89 \\ 90 - 99 & 4 & 90 & 99 \end{array}$

Step 2.2

The class midpoint is the lower class limit plus the upper class limit divided by $2$ .

$\begin{array}{c} Class & Frequency (f) & Lower Limits & Upper Limits & Midpoint (M) \\ 40 - 49 & 1 & 40 & 49 & \frac{40 + 49}{2} \\ 50 - 59 & 2 & 50 & 59 & \frac{50 + 59}{2} \\ 60 - 69 & 3 & 60 & 69 & \frac{60 + 69}{2} \\ 70 - 79 & 4 & 70 & 79 & \frac{70 + 79}{2} \\ 80 - 89 & 6 & 80 & 89 & \frac{80 + 89}{2} \\ 90 - 99 & 4 & 90 & 99 & \frac{90 + 99}{2} \end{array}$

Step 2.3

Simplify all the midpoint column.

$\begin{array}{c} Class & Frequency (f) & Lower Limits & Upper Limits & Midpoint (M) \\ 40 - 49 & 1 & 40 & 49 & 44.5 \\ 50 - 59 & 2 & 50 & 59 & 54.5 \\ 60 - 69 & 3 & 60 & 69 & 64.5 \\ 70 - 79 & 4 & 70 & 79 & 74.5 \\ 80 - 89 & 6 & 80 & 89 & 84.5 \\ 90 - 99 & 4 & 90 & 99 & 94.5 \end{array}$

Step 2.4

Add the midpoints column to the original table.

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) \\ 40 - 49 & 1 & 44.5 \\ 50 - 59 & 2 & 54.5 \\ 60 - 69 & 3 & 64.5 \\ 70 - 79 & 4 & 74.5 \\ 80 - 89 & 6 & 84.5 \\ 90 - 99 & 4 & 94.5 \end{array}$

Step 3

Calculate the square of each group midpoint $M^{2}$ .

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} \\ 40 - 49 & 1 & 44.5 & {44.5}^{2} \\ 50 - 59 & 2 & 54.5 & {54.5}^{2} \\ 60 - 69 & 3 & 64.5 & {64.5}^{2} \\ 70 - 79 & 4 & 74.5 & {74.5}^{2} \\ 80 - 89 & 6 & 84.5 & {84.5}^{2} \\ 90 - 99 & 4 & 94.5 & {94.5}^{2} \end{array}$

Step 4

Simplify the $M^{2}$ column.

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} \\ 40 - 49 & 1 & 44.5 & 1980.25 \\ 50 - 59 & 2 & 54.5 & 2970.25 \\ 60 - 69 & 3 & 64.5 & 4160.25 \\ 70 - 79 & 4 & 74.5 & 5550.25 \\ 80 - 89 & 6 & 84.5 & 7140.25 \\ 90 - 99 & 4 & 94.5 & 8930.25 \end{array}$

Step 5

Multiply each midpoint squared by its frequency $f$ .

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} & f \cdot M^{2} \\ 40 - 49 & 1 & 44.5 & 1980.25 & 1 \cdot 1980.25 \\ 50 - 59 & 2 & 54.5 & 2970.25 & 2 \cdot 2970.25 \\ 60 - 69 & 3 & 64.5 & 4160.25 & 3 \cdot 4160.25 \\ 70 - 79 & 4 & 74.5 & 5550.25 & 4 \cdot 5550.25 \\ 80 - 89 & 6 & 84.5 & 7140.25 & 6 \cdot 7140.25 \\ 90 - 99 & 4 & 94.5 & 8930.25 & 4 \cdot 8930.25 \end{array}$

Step 6

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

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} & f \cdot M^{2} \\ 40 - 49 & 1 & 44.5 & 1980.25 & 1980.25 \\ 50 - 59 & 2 & 54.5 & 2970.25 & 5940.5 \\ 60 - 69 & 3 & 64.5 & 4160.25 & 12480.75 \\ 70 - 79 & 4 & 74.5 & 5550.25 & 22201 \\ 80 - 89 & 6 & 84.5 & 7140.25 & 42841.5 \\ 90 - 99 & 4 & 94.5 & 8930.25 & 35721 \end{array}$

Step 7

Find the sum of all frequencies. In this case, the sum of all frequencies is $n = 1, 2, 3, 4, 6, 4 = 20$ .

$\sum_{}^{} f = n = 20$

Step 8

Find the sum of $f \cdot M^{2}$ column. In this case, $1980.25 + 5940.5 + 12480.75 + 22201 + 42841.5 + 35721 = 121165$ .

$\sum_{}^{} f \cdot M^{2} = 121165$

Step 9

Find the mean $μ$ .

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Step 9.1

Reorder the classes with their related frequencies (ƒ) in an ascending order (lowest number to highest), which is the most common.

$\begin{array}{c} Class & Frequency (f) \\ 40 - 49 & 1 \\ 50 - 59 & 2 \\ 60 - 69 & 3 \\ 70 - 79 & 4 \\ 80 - 89 & 6 \\ 90 - 99 & 4 \end{array}$

Step 9.2

Find the midpoint $M$ for each class.

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) \\ 40 - 49 & 1 & 44.5 \\ 50 - 59 & 2 & 54.5 \\ 60 - 69 & 3 & 64.5 \\ 70 - 79 & 4 & 74.5 \\ 80 - 89 & 6 & 84.5 \\ 90 - 99 & 4 & 94.5 \end{array}$

Step 9.3

Multiply the frequency of each class by the class midpoint.

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & f \cdot M \\ 40 - 49 & 1 & 44.5 & 1 \cdot 44.5 \\ 50 - 59 & 2 & 54.5 & 2 \cdot 54.5 \\ 60 - 69 & 3 & 64.5 & 3 \cdot 64.5 \\ 70 - 79 & 4 & 74.5 & 4 \cdot 74.5 \\ 80 - 89 & 6 & 84.5 & 6 \cdot 84.5 \\ 90 - 99 & 4 & 94.5 & 4 \cdot 94.5 \end{array}$

Step 9.4

Simplify the $f \cdot M$ column.

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & f \cdot M \\ 40 - 49 & 1 & 44.5 & 44.5 \\ 50 - 59 & 2 & 54.5 & 109 \\ 60 - 69 & 3 & 64.5 & 193.5 \\ 70 - 79 & 4 & 74.5 & 298 \\ 80 - 89 & 6 & 84.5 & 507 \\ 90 - 99 & 4 & 94.5 & 378 \end{array}$

Step 9.5

Add the values in the $f \cdot M$ column.

$44.5 + 109 + 193.5 + 298 + 507 + 378 = 1530$

Step 9.6

Add the values in the frequency column.

$n = 1 + 2 + 3 + 4 + 6 + 4 = 20$

Step 9.7

The mean (mu) is the sum of $f \cdot M$ divided by $n$ , which is the sum of frequencies.

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

Step 9.8

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

$μ = \frac{1530}{20}$

Step 9.9

Simplify the right side of $μ = \frac{1530}{20}$ .

$76.5$

Step 10

The equation for the standard deviation is $S^{2} = \frac{\sum_{}^{} f \cdot M^{2} - n {(μ)}^{2}}{n - 1}$ .

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

Step 11

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

$S^{2} = \frac{121165 - 20 {(76.5)}^{2}}{20 - 1}$

Step 12

Simplify the right side of $S^{2} = \frac{121165 - 20 {(76.5)}^{2}}{20 - 1}$ to get the variance $S^{2} = 216.84210526$ .

$216.84210526$

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