document.write( "Question 1184200: The central statistical office publishes a variety of information on the Zambian
\n" ); document.write( "population. The following table shows this.
\n" ); document.write( "
\n" ); document.write( "AGE FREQUENCY
\n" ); document.write( "40 - 49 7
\n" ); document.write( "50 - 59 9
\n" ); document.write( "60 - 69 10
\n" ); document.write( "70 - 79 6
\n" ); document.write( "80 - 89 13
\n" ); document.write( "90 - 99 10
\n" ); document.write( "100 - 109 12
\n" ); document.write( "110 - 119 7
\n" ); document.write( "Compute the following:
\n" ); document.write( "i. The mean
\n" ); document.write( "ii. The median
\n" ); document.write( "iii. The mode
\n" ); document.write( "iv. The quartile deviation
\n" ); document.write( "v. The mean deviation
\n" ); document.write( "vi. The coefficient of variation \n" ); document.write( "
Algebra.Com's Answer #849869 by CPhill(1987)\"\" \"About 
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Here's how to calculate the requested statistics for the Zambian population age data:\r
\n" ); document.write( "\n" ); document.write( "**1. Calculate Midpoints of Age Intervals:**\r
\n" ); document.write( "\n" ); document.write( "Since we have grouped data, we need to find the midpoint of each age interval to represent the ages in that group.\r
\n" ); document.write( "\n" ); document.write( "| Age Interval | Frequency (f) | Midpoint (x) |
\n" ); document.write( "|---|---|---|
\n" ); document.write( "| 40 - 49 | 7 | 44.5 |
\n" ); document.write( "| 50 - 59 | 9 | 54.5 |
\n" ); document.write( "| 60 - 69 | 10 | 64.5 |
\n" ); document.write( "| 70 - 79 | 6 | 74.5 |
\n" ); document.write( "| 80 - 89 | 13 | 84.5 |
\n" ); document.write( "| 90 - 99 | 10 | 94.5 |
\n" ); document.write( "| 100 - 109 | 12 | 104.5 |
\n" ); document.write( "| 110 - 119 | 7 | 114.5 |\r
\n" ); document.write( "\n" ); document.write( "**2. Calculate f * x:**\r
\n" ); document.write( "\n" ); document.write( "Multiply the frequency of each group by its midpoint.\r
\n" ); document.write( "\n" ); document.write( "| Age Interval | Frequency (f) | Midpoint (x) | f * x |
\n" ); document.write( "|---|---|---|---|
\n" ); document.write( "| 40 - 49 | 7 | 44.5 | 311.5 |
\n" ); document.write( "| 50 - 59 | 9 | 54.5 | 490.5 |
\n" ); document.write( "| 60 - 69 | 10 | 64.5 | 645 |
\n" ); document.write( "| 70 - 79 | 6 | 74.5 | 447 |
\n" ); document.write( "| 80 - 89 | 13 | 84.5 | 1098.5 |
\n" ); document.write( "| 90 - 99 | 10 | 94.5 | 945 |
\n" ); document.write( "| 100 - 109 | 12 | 104.5 | 1254 |
\n" ); document.write( "| 110 - 119 | 7 | 114.5 | 801.5 |
\n" ); document.write( "| **Totals** | **74** | | **6000** |\r
\n" ); document.write( "\n" ); document.write( "**i. Mean:**\r
\n" ); document.write( "\n" ); document.write( "Mean (μ) = Σ(f * x) / Σf = 6000 / 74 ≈ 81.08\r
\n" ); document.write( "\n" ); document.write( "**ii. Median:**\r
\n" ); document.write( "\n" ); document.write( "* Total frequency (N) = 74
\n" ); document.write( "* 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.
\n" ); document.write( "* The cumulative frequencies are 7, 16, 26, 32, 45, 55, 67, 74.
\n" ); document.write( "* Both the 37th and 38th values fall in the 80-89 age group.
\n" ); document.write( "* Median = Midpoint of the 80-89 group = 84.5\r
\n" ); document.write( "\n" ); document.write( "**iii. Mode:**\r
\n" ); document.write( "\n" ); document.write( "The mode is the age group with the highest frequency. The 80-89 age group has the highest frequency (13).\r
\n" ); document.write( "\n" ); document.write( "Mode = Midpoint of the 80-89 group = 84.5\r
\n" ); document.write( "\n" ); document.write( "**iv. Quartile Deviation:**\r
\n" ); document.write( "\n" ); document.write( "* Q1 (First Quartile): The value at (N/4) = 74/4 = 18.5. This falls in the 60-69 group.
\n" ); document.write( " * Q1 = 60 + [(18.5 - 16)/10]*10=62.5
\n" ); document.write( "* Q3 (Third Quartile): The value at (3N/4) = 3 * 74/4 = 55.5. This falls in the 90-99 group.
\n" ); document.write( " * Q3 = 90 + [(55.5 - 55)/10]*10=90.5
\n" ); document.write( "* Quartile Deviation = (Q3 - Q1) / 2 = (90.5 - 62.5) / 2 = 28/2 = 14\r
\n" ); document.write( "\n" ); document.write( "**v. Mean Deviation:**\r
\n" ); document.write( "\n" ); document.write( "1. Calculate the absolute deviations |x - μ| for each midpoint.
\n" ); document.write( "2. Multiply each absolute deviation by its frequency (f * |x - μ|).
\n" ); document.write( "3. Sum the results.
\n" ); document.write( "4. Divide by the total frequency (N).\r
\n" ); document.write( "\n" ); document.write( "| Age Interval | f | x | |x-μ| | f*|x-μ| |
\n" ); document.write( "|---|---|---|---|---|---|
\n" ); document.write( "| 40-49 | 7 | 44.5 | |36.58| |256.06|
\n" ); document.write( "| 50-59 | 9 | 54.5 | |26.58| |239.22|
\n" ); document.write( "| 60-69 | 10 | 64.5 | |16.58| |165.8|
\n" ); document.write( "| 70-79 | 6 | 74.5 | |6.58| |39.48|
\n" ); document.write( "| 80-89 | 13 | 84.5 | |3.42| |44.46|
\n" ); document.write( "| 90-99 | 10 | 94.5 | |13.42| |134.2|
\n" ); document.write( "| 100-109 | 12 | 104.5 | |23.42| |281.04|
\n" ); document.write( "| 110-119 | 7 | 114.5 | |33.42| |233.94|
\n" ); document.write( "| **Totals** | 74 | | | | 1394.7 |\r
\n" ); document.write( "\n" ); document.write( "Mean Deviation = 1394.7 / 74 ≈ 18.85\r
\n" ); document.write( "\n" ); document.write( "**vi. Coefficient of Variation:**\r
\n" ); document.write( "\n" ); document.write( "Coefficient of Variation (CV) = (Standard Deviation / Mean) * 100\r
\n" ); document.write( "\n" ); document.write( "We need to calculate the standard deviation first.\r
\n" ); document.write( "\n" ); document.write( "Standard Deviation (σ) = sqrt[Σf(x-μ)²/N]\r
\n" ); document.write( "\n" ); document.write( "= sqrt[Σf(x-μ)²/N]\r
\n" ); document.write( "\n" ); document.write( "= sqrt[10695.82/74] = sqrt[144.54] = 12.02\r
\n" ); document.write( "\n" ); document.write( "CV = (12.02 / 81.08) * 100 ≈ 14.82%
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