Calculate the standard deviation for the given frequency distribution of study hours.

Question

ScanSolve AI Tutor · Accepted Answer

To calculate the standard deviation from a frequency distribution, we use the formula:

\sigma = \sqrt{\frac{\sum fx^2}{\sum f} - \left(\frac{\sum fx}{\sum f}\right)^2}

where $x$ represents the number of hours, $f$ represents the number of students (frequency), $\sum f$ is the total number of students, and $\sum fx$ is the sum of (hours $\times$ frequency).

First, let's organize the data and calculate the necessary sums:

Number of hours ( $x$ )	Number of students ( $f$ )	$fx$	$x^2$	$fx^2$
0	1	0	0	0
1	3	3	1	3
2	3	6	4	12
3	3	9	9	27
4	5	20	16	80
5	6	30	25	150
6	2	12	36	72
7	1	7	49	49
Total	$\sum f = 24$	$\sum fx = 87$		$\sum fx^2 = 393$

Calculate the total number of students ( $\sum f$ ).
$\sum f = 1 + 3 + 3 + 3 + 5 + 6 + 2 + 1 = 24$
Calculate the sum of (hours $\times$ frequency) ( $\sum fx$ ).
$\sum fx = (0 \times 1) + (1 \times 3) + (2 \times 3) + (3 \times 3) + (4 \times 5) + (5 \times 6) + (6 \times 2) + (7 \times 1)$ $\sum fx = 0 + 3 + 6 + 9 + 20 + 30 + 12 + 7 = 87$
Calculate the sum of (hours squared $\times$ frequency) ( $\sum fx^2$ ).
$\sum fx^2 = (0^2 \times 1) + (1^2 \times 3) + (2^2 \times 3) + (3^2 \times 3) + (4^2 \times 5) + (5^2 \times 6) + (6^2 \times 2) + (7^2 \times 1)$ $\sum fx^2 = (0 \times 1) + (1 \times 3) + (4 \times 3) + (9 \times 3) + (16 \times 5) + (25 \times 6) + (36 \times 2) + (49 \times 1)$ $\sum fx^2 = 0 + 3 + 12 + 27 + 80 + 150 + 72 + 49 = 393$
Calculate the mean ( $\bar{x}$ ).
$\bar{x} = \frac{\sum fx}{\sum f} = \frac{87}{24} = 3.625$
Substitute the values into the standard deviation formula.
$\sigma = \sqrt{\frac{\sum fx^2}{\sum f} - \left(\frac{\sum fx}{\sum f}\right)^2}$ $\sigma = \sqrt{\frac{393}{24} - \left(\frac{87}{24}\right)^2}$ $\sigma = \sqrt{16.375 - (3.625)^2}$ $\sigma = \sqrt{16.375 - 13.140625}$ $\sigma = \sqrt{3.234375}$ $\sigma \approx 1.7984368$
Rounding to three decimal places:
$\sigma \approx 1.798$
The standard deviation is 1.798.

Calculate the standard deviation for the given frequency distribution of study hours.

Answer

Step-by-step solution

Calculate the total number of students ( $\sum f$ ).

Calculate the sum of (hours $\times$ frequency) ( $\sum fx$ ).

Calculate the sum of (hours squared $\times$ frequency) ( $\sum fx^2$ ).

Calculate the mean ( $\bar{x}$ ).

Substitute the values into the standard deviation formula.

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Calculate the standard deviation for the given frequency distribution of study hours.

Answer

Step-by-step solution

Calculate the total number of students (∑f\sum f∑f).

Calculate the sum of (hours ×\times× frequency) (∑fx\sum fx∑fx).

Calculate the sum of (hours squared ×\times× frequency) (∑fx2\sum fx^2∑fx2).

Calculate the mean (xˉ\bar{x}xˉ).

Substitute the values into the standard deviation formula.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Calculate the total number of students ( $\sum f$ ).

Calculate the sum of (hours $\times$ frequency) ( $\sum fx$ ).

Calculate the sum of (hours squared $\times$ frequency) ( $\sum fx^2$ ).

Calculate the mean ( $\bar{x}$ ).