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Calculate the mean, variance, standard deviation, and mode for the given grouped frequency distribution.

Mathematics

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Answers

3 parts
a

15.6 minutes\text{15.6 minutes}

b

76.4 minutes2\text{76.4 minutes}^2

c

8.7 minutes\text{8.7 minutes}

Step-by-step solution

12 steps

To compute the mean, variance, standard deviation, and mode for the given grouped frequency distribution, we first need to determine the class mark (xx) for each interval and then calculate the sums required for the formulas.

Here is the table with the necessary calculations:

Class IntervalFrequency (f)Class Mark (x)fxfxx2x^2fx2fx^2
0 - 592.522.56.2556.25
6 - 11168.5136.072.251156.00
12 - 171614.5232.0210.253364.00
18 - 231820.5369.0420.257564.50
24 - 29526.5132.5702.253511.25
30 - 35532.5162.51056.255281.25
36 - 41138.538.51482.251482.25
Total∑f=70\sum f = 70∑fx=1093.0\sum fx = 1093.0∑fx2=22415.5\sum fx^2 = 22415.5

Part a· 3 steps

Mean

  1. Use the formula for the mean of a grouped frequency distribution.

    xˉ=∑fx∑f\bar{x} = \frac{\sum fx}{\sum f}
  2. Substitute the calculated sums.

    xˉ=109370\bar{x} = \frac{1093}{70}
  3. Calculate the mean and round to one decimal place.

    xˉ=15.614...≈15.6\bar{x} = 15.614... \approx 15.6

    The mean is 15.6 minutes\boxed{\text{15.6 minutes}}.

Part b

76.4 minutes2\text{76.4 minutes}^2

Variance

  1. Use the formula for the variance of a grouped frequency distribution.

    s2=∑fx2∑f−(∑fx∑f)2s^2 = \frac{\sum fx^2}{\sum f} - \left(\frac{\sum fx}{\sum f}\right)^2
  2. Substitute the calculated sums.

    s2=22415.570−(109370)2s^2 = \frac{22415.5}{70} - \left(\frac{1093}{70}\right)^2
  3. Calculate the terms.

    s2=320.2214...−(15.614...)2s^2 = 320.2214... - (15.614...)^2 s2=320.2214...−243.7906...s^2 = 320.2214... - 243.7906...
  4. Calculate the variance and round to one decimal place.

    s2=76.4308...≈76.4s^2 = 76.4308... \approx 76.4

    The variance is 76.4 minutes2\boxed{\text{76.4 minutes}^2}.

Part c

8.7 minutes\text{8.7 minutes}

Standard Deviation

  1. Use the formula for standard deviation, which is the square root of the variance.

    s=s2s = \sqrt{s^2}
  2. Substitute the calculated variance.

    s=76.4308...s = \sqrt{76.4308...}
  3. Calculate the standard deviation and round to one decimal place.

    s=8.742...≈8.7s = 8.742... \approx 8.7

    The standard deviation is 8.7 minutes\boxed{\text{8.7 minutes}}.

Part d· 2 steps

Mode

  1. Identify the modal class, which is the class with the highest frequency.

    The highest frequency is 18, corresponding to the class interval 18 - 23.

  2. Identify the

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