Calculate the mean from the following table.
Scores
Frequencies
0-10
2
10-20
4
20-30
12
30-40
21
40-50
6
50-60
3
60-70
2

Question

Calculate the mean from the following table.ScoresFrequencies0-10210-20420-301230-402140-50650-60360-702

Prepp Team · Accepted Answer

Calculating the Mean from a Frequency Table

To find the mean (average) from a frequency distribution table like the one provided, where data is grouped into class intervals, we follow these steps:

Find the midpoint of each class interval. The midpoint is calculated as (Lower limit + Upper limit) / 2.
Multiply the midpoint of each class interval by its corresponding frequency. This gives us the product $f_i \times x_i$, where $f_i$ is the frequency and $x_i$ is the midpoint.
Sum up all the products ($f_i \times x_i$). This is denoted as $\sum f_ix_i$.
Sum up all the frequencies. This is denoted as $\sum f_i$.
Calculate the mean using the formula:

$$\text{Mean} = \frac{\sum f_ix_i}{\sum f_i}$$

Step-by-Step Mean Calculation for the Given Frequency Table

Let's apply these steps to the given table of scores and frequencies:

Scores (Class Interval)	Frequency ($f_i$)	Midpoint ($x_i$)	$f_ix_i$
0-10	2	$$\frac{0+10}{2} = 5$$	$$2 \times 5 = 10$$
10-20	4	$$\frac{10+20}{2} = 15$$	$$4 \times 15 = 60$$
20-30	12	$$\frac{20+30}{2} = 25$$	$$12 \times 25 = 300$$
30-40	21	$$\frac{30+40}{2} = 35$$	$$21 \times 35 = 735$$
40-50	6	$$\frac{40+50}{2} = 45$$	$$6 \times 45 = 270$$
50-60	3	$$\frac{50+60}{2} = 55$$	$$3 \times 55 = 165$$
60-70	2	$$\frac{60+70}{2} = 65$$	$$2 \times 65 = 130$$

Now, we calculate the sums:

Sum of frequencies ($\sum f_i$): $$2 + 4 + 12 + 21 + 6 + 3 + 2 = 50$$
Sum of products ($f_ix_i$): $$10 + 60 + 300 + 735 + 270 + 165 + 130 = 1670$$

Calculating the Mean Score

Using the formula for the mean of grouped data:

$$\text{Mean} = \frac{\sum f_ix_i}{\sum f_i}$$

Substituting the calculated values:

$$\text{Mean} = \frac{1670}{50}$$

$$\text{Mean} = 33.4$$

Therefore, the mean score calculated from the given frequency table is 33.4.

Revision Table: Key Terms for Frequency Distributions

Term	Definition
Frequency	The number of times a particular value or range of values (class interval) appears in a dataset.
Frequency Distribution Table	A table that lists scores or class intervals and their corresponding frequencies.
Class Interval	A range into which data is grouped in a frequency distribution. E.g., 0-10, 10-20.
Midpoint (Class Mark)	The average of the upper and lower limits of a class interval. Used to represent the interval in mean calculation.
Mean	The arithmetic average of a dataset. For grouped data, it's the sum of ($f_ix_i$) divided by the sum of frequencies.

Additional Information: Measures of Central Tendency

The mean is one of the key measures of central tendency used in statistics to represent the center or typical value of a dataset. Besides the mean, other important measures include:

Median: The middle value in a dataset when arranged in order. For grouped data, it requires finding the median class.
Mode: The value that appears most frequently in a dataset. For grouped data, it requires finding the modal class.

These measures provide different perspectives on the center of the data distribution. The mean is sensitive to extreme values, while the median is not.