Question 1184200: The central statistical office publishes a variety of information on the Zambian
population. The following table shows this.
AGE FREQUENCY
40 - 49 7
50 - 59 9
60 - 69 10
70 - 79 6
80 - 89 13
90 - 99 10
100 - 109 12
110 - 119 7
Compute the following:
i. The mean
ii. The median
iii. The mode
iv. The quartile deviation
v. The mean deviation
vi. The coefficient of variation
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the requested statistics for the Zambian population age data:
**1. Calculate Midpoints of Age Intervals:**
Since we have grouped data, we need to find the midpoint of each age interval to represent the ages in that group.
| Age Interval | Frequency (f) | Midpoint (x) |
|---|---|---|
| 40 - 49 | 7 | 44.5 |
| 50 - 59 | 9 | 54.5 |
| 60 - 69 | 10 | 64.5 |
| 70 - 79 | 6 | 74.5 |
| 80 - 89 | 13 | 84.5 |
| 90 - 99 | 10 | 94.5 |
| 100 - 109 | 12 | 104.5 |
| 110 - 119 | 7 | 114.5 |
**2. Calculate f * x:**
Multiply the frequency of each group by its midpoint.
| Age Interval | Frequency (f) | Midpoint (x) | f * x |
|---|---|---|---|
| 40 - 49 | 7 | 44.5 | 311.5 |
| 50 - 59 | 9 | 54.5 | 490.5 |
| 60 - 69 | 10 | 64.5 | 645 |
| 70 - 79 | 6 | 74.5 | 447 |
| 80 - 89 | 13 | 84.5 | 1098.5 |
| 90 - 99 | 10 | 94.5 | 945 |
| 100 - 109 | 12 | 104.5 | 1254 |
| 110 - 119 | 7 | 114.5 | 801.5 |
| **Totals** | **74** | | **6000** |
**i. Mean:**
Mean (μ) = Σ(f * x) / Σf = 6000 / 74 ≈ 81.08
**ii. Median:**
* Total frequency (N) = 74
* The median is the average of the (N/2)th and ((N/2) + 1)th values. Since N is even, it's the average of the 37th and 38th values.
* The cumulative frequencies are 7, 16, 26, 32, 45, 55, 67, 74.
* Both the 37th and 38th values fall in the 80-89 age group.
* Median = Midpoint of the 80-89 group = 84.5
**iii. Mode:**
The mode is the age group with the highest frequency. The 80-89 age group has the highest frequency (13).
Mode = Midpoint of the 80-89 group = 84.5
**iv. Quartile Deviation:**
* Q1 (First Quartile): The value at (N/4) = 74/4 = 18.5. This falls in the 60-69 group.
* Q1 = 60 + [(18.5 - 16)/10]*10=62.5
* Q3 (Third Quartile): The value at (3N/4) = 3 * 74/4 = 55.5. This falls in the 90-99 group.
* Q3 = 90 + [(55.5 - 55)/10]*10=90.5
* Quartile Deviation = (Q3 - Q1) / 2 = (90.5 - 62.5) / 2 = 28/2 = 14
**v. Mean Deviation:**
1. Calculate the absolute deviations |x - μ| for each midpoint.
2. Multiply each absolute deviation by its frequency (f * |x - μ|).
3. Sum the results.
4. Divide by the total frequency (N).
| Age Interval | f | x | |x-μ| | f*|x-μ| |
|---|---|---|---|---|---|
| 40-49 | 7 | 44.5 | |36.58| |256.06|
| 50-59 | 9 | 54.5 | |26.58| |239.22|
| 60-69 | 10 | 64.5 | |16.58| |165.8|
| 70-79 | 6 | 74.5 | |6.58| |39.48|
| 80-89 | 13 | 84.5 | |3.42| |44.46|
| 90-99 | 10 | 94.5 | |13.42| |134.2|
| 100-109 | 12 | 104.5 | |23.42| |281.04|
| 110-119 | 7 | 114.5 | |33.42| |233.94|
| **Totals** | 74 | | | | 1394.7 |
Mean Deviation = 1394.7 / 74 ≈ 18.85
**vi. Coefficient of Variation:**
Coefficient of Variation (CV) = (Standard Deviation / Mean) * 100
We need to calculate the standard deviation first.
Standard Deviation (σ) = sqrt[Σf(x-μ)²/N]
= sqrt[Σf(x-μ)²/N]
= sqrt[10695.82/74] = sqrt[144.54] = 12.02
CV = (12.02 / 81.08) * 100 ≈ 14.82%
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