Questions: Question 2 [24] The market price per share of 1000 selected JSE-listed services companies is shown in the distribution table below: Market Price Per Share Frequency 2.5-<5.0 41 5.0-<7.5 98 7.5-<10.0 101 10.0-<12.5 160 12.5-<15.0 210 15.0-<17.5 140 17.5-<20.0 88 20.0-<22.5 94 22.5-<25.0 68 2.1. Calculate the following measures for the data: 2.1.1. Mean (3) 2.1.2. Median (3) 2.1.3. Standard deviation (5)

Solution Steps

To solve the given problem, we need to calculate the mean, median, and standard deviation of the market price per share data provided in the frequency distribution table.

• Calculate the midpoint for each class interval.
• Multiply each midpoint by its corresponding frequency to get the total for each class.
• Sum these totals and divide by the total frequency to find the mean.

• Determine the cumulative frequency for each class interval.
• Find the class interval where the cumulative frequency exceeds half of the total frequency.
• Use the median formula for grouped data to calculate the median.

• Calculate the variance by finding the squared difference between each class midpoint and the mean, weighted by the frequency.
• Sum these values, divide by the total frequency, and take the square root to find the standard deviation.

To find the mean of the data, we use the formula for the mean of grouped data:

\[ \text{Mean} = \frac{\sum (f_i \cdot x_i)}{\sum f_i} \]

where \( f_i \) is the frequency and \( x_i \) is the midpoint of each class interval. Given the midpoints and frequencies:

\[ \text{Mean} = \frac{41 \cdot 3.75 + 98 \cdot 6.25 + 101 \cdot 8.75 + 160 \cdot 11.25 + 210 \cdot 13.75 + 140 \cdot 16.25 + 88 \cdot 18.75 + 94 \cdot 21.25 + 68 \cdot 23.75}{1000} = 13.875 \]

The median is found using the formula for the median of grouped data:

\[ \text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right) \cdot h \]

where:

• \( L = 12.5 \) is the lower boundary of the median class,
• \( N = 1000 \) is the total frequency,
• \( F = 400 \) is the cumulative frequency before the median class,
• \( f = 210 \) is the frequency of the median class,
• \( h = 2.5 \) is the class width.

\[ \text{Median} = 12.5 + \left(\frac{500 - 400}{210}\right) \cdot 2.5 = 13.6905 \]

The standard deviation is calculated using the formula for the standard deviation of grouped data:

\[ \sigma = \sqrt{\frac{\sum f_i (x_i - \text{Mean})^2}{\sum f_i}} \]

Given the variance:

\[ \text{Variance} = 28.284375 \]

The standard deviation is:

\[ \sigma = \sqrt{28.284375} = 5.318 \]

Final Answer