Construct a grouped frequency distribution table and calculate the mean mark and standard deviation given a…

Question

Construct a grouped frequency distribution table and calculate the mean mark and standard deviation given a set of data and an assumed mean.

ScanSolve AI Tutor · Accepted Answer

here's the solution for question 12:

a) Construct a grouped frequency distribution table using the class interval 20-29, 30-39, 40-49...

Define class intervals and tally the given marks.
The marks are:
29 52 37 79 63 48 65 46 54 32 51 66 74 24 35 61 58 39 49 53
45 58 56 38 59 28 50 49 67 56 37 45 79 68 43 66 26 47 55 71

The class intervals are:
20-29
30-39
40-49
50-59
60-69
70-79
Count the frequency for each class interval.
- 20-29: 29, 24, 28, 26 (4 marks)
- 30-39: 37, 32, 35, 39, 38, 37 (6 marks)
- 40-49: 48, 46, 49, 45, 49, 43, 45, 47 (8 marks)
- 50-59: 52, 54, 51, 53, 58, 58, 56, 59, 50, 56, 55 (11 marks)
- 60-69: 63, 65, 66, 61, 67, 68, 66 (7 marks)
- 70-79: 79, 74, 79, 71 (4 marks)
Construct the frequency distribution table.
$\begin{array}{|c|c|c|} \hline \textbf{Class Interval} & \textbf{Tally} & \textbf{Frequency (f)} \\ \hline 20-29 & |||| & 4 \\ 30-39 & ||||| | & 6 \\ 40-49 & ||||| ||| & 8 \\ 50-59 & ||||| ||||| | & 11 \\ 60-69 & ||||| || & 7 \\ 70-79 & |||| & 4 \\ \hline \textbf{Total} & & \textbf{40} \\ \hline \end{array}$
b) Using an assumed mean of 54.5, calculate, correct to the nearest whole number, the:

(i) mean mark;

(ii) standard deviation of the distribution.
Create an extended frequency table to calculate class marks ( $x$ ), deviations ( $d$ ), $fd$ , and $fd^2$ .
The assumed mean $A = 54.5$ .
The class mark $x$ is the midpoint of each interval. For example, for 20-29, $x = \frac{20+29}{2} = 24.5$ .
The deviation $d = x - A$ .
$\begin{array}{|c|c|c|c|c|c|c|} \hline \textbf{Class Interval} & \textbf{f} & \textbf{x} & \textbf{d = x - 54.5} & \textbf{fd} & \textbf{d}^2 & \textbf{fd}^2 \\ \hline 20-29 & 4 & 24.5 & -30 & -120 & 900 & 3600 \\ 30-39 & 6 & 34.5 & -20 & -120 & 400 & 2400 \\ 40-49 & 8 & 44.5 & -10 & -80 & 100 & 800 \\ 50-59 & 11 & 54.5 & 0 & 0 & 0 & 0 \\ 60-69 & 7 & 64.5 & 10 & 70 & 100 & 700 \\ 70-79 & 4 & 74.5 & 20 & 80 & 400 & 1600 \\ \hline \textbf{Total} & \sum f = 40 & & & \sum fd = -170 & & \sum fd^2 = 9100 \\ \hline \end{array}$
(i) Calculate the mean mark.
The formula for the mean using an assumed mean is $\bar{x} = A + \frac{\sum fd}{\sum f}$ .
$\bar{x} = 54.5 + \frac{-170}{40}$ $\bar{x} = 54.5 - 4.25$ $\bar{x} = 50.25$
Correct to the nearest whole number, the mean mark is $\boxed{\text{50}}$ .

(ii) Calculate the standard deviation of the distribution.
The formula for standard deviation using an assumed mean is $\sigma = \sqrt{\frac{\sum fd^2}{\sum f} - \left(\frac{\sum fd}{\sum f}\right)^2}$ .
$\sigma = \sqrt{\frac{9100}{40} - \left(\frac{-170}{40}\right)^2}$ $\sigma = \sqrt{227.5 - (-4.25)^2}$ $\sigma = \sqrt{227.5 - 18.0625}$ $\sigma = \sqrt{209.4375}$ $\sigma \approx 14.472$
Correct to the nearest whole number, the standard deviation is $\boxed{\text{14}}$ .

Construct a grouped frequency distribution table and calculate the mean mark and standard deviation given a set of data and an assumed mean.

Step-by-step solution

Define class intervals and tally the given marks.

Count the frequency for each class interval.

Construct the frequency distribution table.

Create an extended frequency table to calculate class marks ( $x$ ), deviations ( $d$ ), $fd$ , and $fd^2$ .

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More Mathematics Questions

Construct a grouped frequency distribution table and calculate the mean mark and standard deviation given a set of data and an assumed mean.

Step-by-step solution

Define class intervals and tally the given marks.

Count the frequency for each class interval.

Construct the frequency distribution table.

Create an extended frequency table to calculate class marks (xxx), deviations (ddd), fdfdfd, and fd2fd^2fd2.

Still stuck on this one?

More Mathematics Questions

Create an extended frequency table to calculate class marks ( $x$ ), deviations ( $d$ ), $fd$ , and $fd^2$ .