Statistics Examples

$\begin{array}{c} Class & Frequency \\ 360 - 369 & 2 \\ 370 - 379 & 3 \\ 380 - 389 & 5 \\ 390 - 399 & 7 \\ 400 - 409 & 5 \\ 410 - 419 & 4 \\ 420 - 429 & 4 \\ 430 - 439 & 1 \end{array}$

Step 1

Find the midpoint

$M$ 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.

$\begin{array}{c} Class & Frequency (f) & Lower Limits & Upper Limits \\ 360 - 369 & 2 & 360 & 369 \\ 370 - 379 & 3 & 370 & 379 \\ 380 - 389 & 5 & 380 & 389 \\ 390 - 399 & 7 & 390 & 399 \\ 400 - 409 & 5 & 400 & 409 \\ 410 - 419 & 4 & 410 & 419 \\ 420 - 429 & 4 & 420 & 429 \\ 430 - 439 & 1 & 430 & 439 \end{array}$

Step 1.2

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

$2$ .

$\begin{array}{c} Class & Frequency (f) & Lower Limits & Upper Limits & Midpoint (M) \\ 360 - 369 & 2 & 360 & 369 & \frac{360 + 369}{2} \\ 370 - 379 & 3 & 370 & 379 & \frac{370 + 379}{2} \\ 380 - 389 & 5 & 380 & 389 & \frac{380 + 389}{2} \\ 390 - 399 & 7 & 390 & 399 & \frac{390 + 399}{2} \\ 400 - 409 & 5 & 400 & 409 & \frac{400 + 409}{2} \\ 410 - 419 & 4 & 410 & 419 & \frac{410 + 419}{2} \\ 420 - 429 & 4 & 420 & 429 & \frac{420 + 429}{2} \\ 430 - 439 & 1 & 430 & 439 & \frac{430 + 439}{2} \end{array}$

Step 1.3

Simplify all the midpoint column.

$\begin{array}{c} Class & Frequency (f) & Lower Limits & Upper Limits & Midpoint (M) \\ 360 - 369 & 2 & 360 & 369 & 364.5 \\ 370 - 379 & 3 & 370 & 379 & 374.5 \\ 380 - 389 & 5 & 380 & 389 & 384.5 \\ 390 - 399 & 7 & 390 & 399 & 394.5 \\ 400 - 409 & 5 & 400 & 409 & 404.5 \\ 410 - 419 & 4 & 410 & 419 & 414.5 \\ 420 - 429 & 4 & 420 & 429 & 424.5 \\ 430 - 439 & 1 & 430 & 439 & 434.5 \end{array}$

Step 1.4

Add the midpoints column to the original table.

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) \\ 360 - 369 & 2 & 364.5 \\ 370 - 379 & 3 & 374.5 \\ 380 - 389 & 5 & 384.5 \\ 390 - 399 & 7 & 394.5 \\ 400 - 409 & 5 & 404.5 \\ 410 - 419 & 4 & 414.5 \\ 420 - 429 & 4 & 424.5 \\ 430 - 439 & 1 & 434.5 \end{array}$

Step 2

Calculate the square of each group midpoint

$M^{2}$ .

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} \\ 360 - 369 & 2 & 364.5 & {364.5}^{2} \\ 370 - 379 & 3 & 374.5 & {374.5}^{2} \\ 380 - 389 & 5 & 384.5 & {384.5}^{2} \\ 390 - 399 & 7 & 394.5 & {394.5}^{2} \\ 400 - 409 & 5 & 404.5 & {404.5}^{2} \\ 410 - 419 & 4 & 414.5 & {414.5}^{2} \\ 420 - 429 & 4 & 424.5 & {424.5}^{2} \\ 430 - 439 & 1 & 434.5 & {434.5}^{2} \end{array}$

Step 3

Simplify the

$M^{2}$ column.

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} \\ 360 - 369 & 2 & 364.5 & 132860.25 \\ 370 - 379 & 3 & 374.5 & 140250.25 \\ 380 - 389 & 5 & 384.5 & 147840.25 \\ 390 - 399 & 7 & 394.5 & 155630.25 \\ 400 - 409 & 5 & 404.5 & 163620.25 \\ 410 - 419 & 4 & 414.5 & 171810.25 \\ 420 - 429 & 4 & 424.5 & 180200.25 \\ 430 - 439 & 1 & 434.5 & 188790.25 \end{array}$

Step 4

Multiply each midpoint squared by its frequency

$f$ .

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} & f \cdot M^{2} \\ 360 - 369 & 2 & 364.5 & 132860.25 & 2 \cdot 132860.25 \\ 370 - 379 & 3 & 374.5 & 140250.25 & 3 \cdot 140250.25 \\ 380 - 389 & 5 & 384.5 & 147840.25 & 5 \cdot 147840.25 \\ 390 - 399 & 7 & 394.5 & 155630.25 & 7 \cdot 155630.25 \\ 400 - 409 & 5 & 404.5 & 163620.25 & 5 \cdot 163620.25 \\ 410 - 419 & 4 & 414.5 & 171810.25 & 4 \cdot 171810.25 \\ 420 - 429 & 4 & 424.5 & 180200.25 & 4 \cdot 180200.25 \\ 430 - 439 & 1 & 434.5 & 188790.25 & 1 \cdot 188790.25 \end{array}$

Step 5

Simplify the

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

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} & f \cdot M^{2} \\ 360 - 369 & 2 & 364.5 & 132860.25 & 265720.5 \\ 370 - 379 & 3 & 374.5 & 140250.25 & 420750.75 \\ 380 - 389 & 5 & 384.5 & 147840.25 & 739201.25 \\ 390 - 399 & 7 & 394.5 & 155630.25 & 1089411.75 \\ 400 - 409 & 5 & 404.5 & 163620.25 & 818101.25 \\ 410 - 419 & 4 & 414.5 & 171810.25 & 687241 \\ 420 - 429 & 4 & 424.5 & 180200.25 & 720801 \\ 430 - 439 & 1 & 434.5 & 188790.25 & 188790.25 \end{array}$

Step 6

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

$n = 2, 3, 5, 7, 5, 4, 4, 1 = 31$ .

$\sum_{}^{} f = n = 31$

Step 7

Find the sum of

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

$265720.5 + 420750.75 + 739201.25 + 1089411.75 + 818101.25 + 687241 + 720801 + 188790.25 = 4930017.75$ .

$\sum_{}^{} f \cdot M^{2} = 4930017.75$

Step 8

Find the mean

$μ$ .

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Step 8.1

Find the midpoint

$M$ for each class.

Step 8.2

Multiply the frequency of each class by the class midpoint.

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & f \cdot M \\ 360 - 369 & 2 & 364.5 & 2 \cdot 364.5 \\ 370 - 379 & 3 & 374.5 & 3 \cdot 374.5 \\ 380 - 389 & 5 & 384.5 & 5 \cdot 384.5 \\ 390 - 399 & 7 & 394.5 & 7 \cdot 394.5 \\ 400 - 409 & 5 & 404.5 & 5 \cdot 404.5 \\ 410 - 419 & 4 & 414.5 & 4 \cdot 414.5 \\ 420 - 429 & 4 & 424.5 & 4 \cdot 424.5 \\ 430 - 439 & 1 & 434.5 & 1 \cdot 434.5 \end{array}$

Step 8.3

Simplify the

$f \cdot M$ column.

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & f \cdot M \\ 360 - 369 & 2 & 364.5 & 729 \\ 370 - 379 & 3 & 374.5 & 1123.5 \\ 380 - 389 & 5 & 384.5 & 1922.5 \\ 390 - 399 & 7 & 394.5 & 2761.5 \\ 400 - 409 & 5 & 404.5 & 2022.5 \\ 410 - 419 & 4 & 414.5 & 1658 \\ 420 - 429 & 4 & 424.5 & 1698 \\ 430 - 439 & 1 & 434.5 & 434.5 \end{array}$

Step 8.4

Add the values in the

$f \cdot M$ column.

$729 + 1123.5 + 1922.5 + 2761.5 + 2022.5 + 1658 + 1698 + 434.5 = 12349.5$

Step 8.5

Add the values in the frequency column.

$n = 2 + 3 + 5 + 7 + 5 + 4 + 4 + 1 = 31$

Step 8.6

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 8.7

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

$μ = \frac{12349.5}{31}$

Step 8.8

Simplify the right side of

$μ = \frac{12349.5}{31}$ .

$398.37096774$

Step 9

The equation for the standard deviation is

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

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

Step 10

Substitute the calculated values into

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

$S^{2} = \frac{4930017.75 - 31 {(398.37096774)}^{2}}{31 - 1}$

Step 11

Simplify the right side of

$S^{2} = \frac{4930017.75 - 31 {(398.37096774)}^{2}}{31 - 1}$ to get the variance

$S^{2} = 344.51612903$ .

$344.51612903$

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