A random sample of 20 people is classified in the following table according to their ages: Age Frequency 15 – 25 2 25 – 35 4 35 – 45 6 45 – 55 5 55 - 65 3 What is the mean age of this group of people?
41.5
Calculate Mean Age from a Frequency Table (Grouped Data)
The problem asks us to find the mean age of a group of 20 people, given in a frequency distribution table. Since the age data is presented in ranges (classes) with associated frequencies, this is a case of calculating the mean for grouped data.
Steps to Calculate Mean for Grouped Data
To find the mean from a frequency distribution table with grouped data, we follow these steps:
- Find the midpoint (also called the class mark) for each age range. The midpoint is the average of the lower and upper limits of the class interval.
- Multiply the frequency of each class by its midpoint ($\text{f} \times \text{x}$), where 'f' is the frequency and 'x' is the midpoint.
- Sum up all the products from step 2 ($\sum \text{f} \times \text{x}$).
- Sum up all the frequencies ($\sum \text{f}$). This sum should equal the total number of observations, which is 20 in this case.
- Use the formula for the mean of grouped data: $\text{Mean} = \frac{\sum \text{f} \times \text{x}}{\sum \text{f}}$.
Applying the Steps to the Age Data
Let's apply these steps to the given frequency table. We will create a new table to include the midpoints and the products of frequency and midpoint.
| Age Range | Frequency (f) | Midpoint (x) | f $\times$ x |
|---|---|---|---|
| 15 – 25 | 2 | (15 + 25) / 2 = 20 | 2 $\times$ 20 = 40 |
| 25 – 35 | 4 | (25 + 35) / 2 = 30 | 4 $\times$ 30 = 120 |
| 35 – 45 | 6 | (35 + 45) / 2 = 40 | 6 $\times$ 40 = 240 |
| 45 – 55 | 5 | (45 + 55) / 2 = 50 | 5 $\times$ 50 = 250 |
| 55 – 65 | 3 | (55 + 65) / 2 = 60 | 3 $\times$ 60 = 180 |
| Total | $\sum \text{f} = 2 + 4 + 6 + 5 + 3 = 20$ | $\sum \text{f} \times \text{x} = 40 + 120 + 240 + 250 + 180 = 830$ |
Calculating the Mean Age
Now we use the formula for the mean of grouped data:
$\text{Mean} = \frac{\sum \text{f} \times \text{x}}{\sum \text{f}}$
Substitute the values we calculated:
$\text{Mean} = \frac{830}{20}$
$\text{Mean} = 41.5$
The mean age of this group of people is 41.5 years.
Revision Table: Key Concepts for Mean Calculation
| Concept | Definition/Purpose | ||
|---|---|---|---|
| Mean | A measure of central tendency, representing the average value of a dataset. | ||
| Frequency Distribution | A table that summarizes how often different values or ranges of values occur in a dataset. | ||
| Grouped Data | Data organized into classes or intervals in a frequency distribution. | Midpoint (Class Mark) | The representative value for a class interval, calculated as (lower limit + upper limit) / 2. Used to estimate the mean for grouped data. |
Additional Information: Why Use Midpoints?
When dealing with grouped data, we do not know the exact value of each observation within a class interval. For instance, in the 15-25 age range, we know there are 2 people, but we don't know their precise ages (e.g., they could be 18 and 22, or 15 and 25). To estimate the mean, we assume that the observations within each class are evenly distributed around the midpoint. Therefore, the midpoint serves as the best representative value for all observations within that particular class when performing calculations like finding the mean.
Important Questions from Measures of Central Tendency
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The mean of the series x 1, x 2… x nis X̅. If x 2 is replaced by λ, then what is the new mean?
The mean of a group of 100 observations was found to be 20. Later it was found that four observations were incorrect, which were recorded as 21, 21, 18 and 20. What is the mean if the incorrect observations are omitted?
