Complete all the problems.
Consider the following three Data Sets A, B and C.
A= {1, 2, 3, 4, 5}
B = {2, 2, 2, 2, 2}
C = {5, 7, 3, 11, 14}
1. Calculate the mean of each data set.
2. Calculate the standard deviation of each data set.
3. Which set has the largest standard deviation?
4. Is it possible to answer question \”c\” without calculations of the standard deviation? The frequency Table is shown below. Number of Children frequency 20 4 40 2 30 6 80 5 60 3
5. CaLculate the mean of the number of children of frequency.
6. Calculate the standard deviation of the number of children of frequency.
The Correct Answer and Explanation is:
Answers
- Calculate the mean of each data set:
- Set A: (1 + 2 + 3 + 4 + 5) ÷ 5 = 3
- Set B: (2 + 2 + 2 + 2 + 2) ÷ 5 = 2
- Set C: (5 + 7 + 3 + 11 + 14) ÷ 5 = 8
- Calculate the standard deviation of each data set:
- Set A: Variance = [(1-3)² + (2-3)² + (3-3)² + (4-3)² + (5-3)²] ÷ 5 = 2 Standard deviation = √2 ≈ 1.41
- Set B: Variance = [(2-2)² + (2-2)² + (2-2)² + (2-2)² + (2-2)²] ÷ 5 = 0 Standard deviation = 0
- Set C: Variance = [(5-8)² + (7-8)² + (3-8)² + (11-8)² + (14-8)²] ÷ 5 = 16 Standard deviation = 4
- Which set has the largest standard deviation? Set C has the largest standard deviation, which is 4.
- Is it possible to answer question 3 without calculating the standard deviation? Yes, by examining the spread of values. Set B has identical values, meaning no variation. Set A has a smaller range between its lowest and highest numbers compared to Set C, which has widely spaced values. Greater spread implies a larger standard deviation.
- Calculate the mean of the number of children in the frequency table: Weighted mean = (20×4 + 40×2 + 30×6 + 80×5 + 60×3) ÷ (4 + 2 + 6 + 5 + 3) = (80 + 80 + 180 + 400 + 180) ÷ 20 = 46
- Calculate the standard deviation of the number of children in the frequency table: Variance = [(20-46)²×4 + (40-46)²×2 + (30-46)²×6 + (80-46)²×5 + (60-46)²×3] ÷ 20 = [676×4 + 36×2 + 256×6 + 1156×5 + 196×3] ÷ 20 = 534 Standard deviation = √534 ≈ 23.11
Explanation
The mean provides a central value representative of a dataset, calculated by summing all values and dividing by the total number of observations. Standard deviation measures data spread from the mean, offering insight into variability. If a dataset has all values equal, as seen in Set B, the standard deviation is zero since there is no variation. Set A has moderate variation with values close to each other, resulting in a lower standard deviation. Set C demonstrates a wide range of values, leading to the highest standard deviation.
For the frequency table, the weighted mean is calculated by multiplying each number of children by its frequency, summing these products, and dividing by the total frequency count. Standard deviation follows similar principles as before, adjusting for frequency by considering squared deviations weighted by occurrences. The calculated standard deviation suggests high dispersion in the number of children per frequency count, indicating significant variation among groups.
By understanding these calculations, one can appreciate the role of statistical measures in evaluating data distribution and variability.
