Calculate the average (mean), variance, and standard deviation for the following data:
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline Age & 18 & 23 & 28 & 33 & 38 & 43 & 48 \\
\hline No & 12 & 16 & 18 & 20 & 10 & 10 & 14 \\
\hline
\end{tabular}
Answer (1)
Calculate the mean: ∑ F re q u e n cy ∑ ( A g e × F re q u e n cy ) = 100 3230 = 32.3 .
Calculate the variance: ∑ F re q u e n cy ∑ F re q u e n cy × ( A g e − M e an ) 2 = 100 9101 = 91.01 .
Calculate the standard deviation: Variance = 91.01 ≈ 9.54 .
The average age is 32.3 , the variance is 91.01 , and the standard deviation is 9.54 .
Explanation
Problem Setup and Formulas We are given a frequency table of ages and asked to calculate the mean, variance, and standard deviation. Let's start by outlining the formulas we'll use.
Formulas for Mean, Variance, and Standard Deviation The mean (average) is calculated as the sum of each age multiplied by its frequency, divided by the total frequency: Mean = ∑ F re q u e n cy ∑ ( A g e × F re q u e n cy ) The variance is calculated as the average of the squared differences from the mean: Variance = ∑ F re q u e n cy ∑ F re q u e n cy × ( A g e − M e an ) 2 The standard deviation is the square root of the variance: Standard Deviation = Variance .
Calculating the Mean First, let's calculate the total frequency: ∑ F re q u e n cy = 12 + 16 + 18 + 20 + 10 + 10 + 14 = 100 Next, we calculate the sum of (Age × Frequency): ∑ ( A g e × F re q u e n cy ) = ( 18 × 12 ) + ( 23 × 16 ) + ( 28 × 18 ) + ( 33 × 20 ) + ( 38 × 10 ) + ( 43 × 10 ) + ( 48 × 14 ) ∑ ( A g e × F re q u e n cy ) = 216 + 368 + 504 + 660 + 380 + 430 + 672 = 3230 Now, we can calculate the mean: Mean = 100 3230 = 32.3 .
Calculating the Variance Now we calculate the variance. We need to find the squared difference between each age and the mean, multiply by the frequency, and sum them up: ∑ F re q u e n cy × ( A g e − M e an ) 2 = 12 × ( 18 − 32.3 ) 2 + 16 × ( 23 − 32.3 ) 2 + 18 × ( 28 − 32.3 ) 2 + 20 × ( 33 − 32.3 ) 2 + 10 × ( 38 − 32.3 ) 2 + 10 × ( 43 − 32.3 ) 2 + 14 × ( 48 − 32.3 ) 2 ∑ F re q u e n cy × ( A g e − M e an ) 2 = 12 × ( − 14.3 ) 2 + 16 × ( − 9.3 ) 2 + 18 × ( − 4.3 ) 2 + 20 × ( 0.7 ) 2 + 10 × ( 5.7 ) 2 + 10 × ( 10.7 ) 2 + 14 × ( 15.7 ) 2 ∑ F re q u e n cy × ( A g e − M e an ) 2 = 12 × 204.49 + 16 × 86.49 + 18 × 18.49 + 20 × 0.49 + 10 × 32.49 + 10 × 114.49 + 14 × 246.49 ∑ F re q u e n cy × ( A g e − M e an ) 2 = 2453.88 + 1383.84 + 332.82 + 9.8 + 324.9 + 1144.9 + 3450.86 = 9101 Now, we can calculate the variance: Variance = 100 9101 = 91.01 .
Calculating the Standard Deviation Finally, we calculate the standard deviation by taking the square root of the variance: Standard Deviation = 91.01 ≈ 9.5399 .
Final Answer The mean age is 32.3, the variance is 91.01, and the standard deviation is approximately 9.54.
Examples
Understanding the distribution of ages in a population can be useful in many real-world scenarios. For example, an insurance company might use this information to assess risk and set premiums. Similarly, a marketing company could use age distribution data to tailor its advertising campaigns to specific age groups. By calculating the mean, variance, and standard deviation, we gain valuable insights into the central tendency and spread of the data, which can inform decision-making in various fields.
