Gabe is the human resource manager for the Advanced Scientific Research Lab. He has to record
the heights (in centimeters) and weights (in pounds) for each of the scientists in the lab.
Height distribution (cm): 178, 163, 174, 186, 154, 167, 167, 181, 159, 165, 177, 191, 158
Weight distribution (lbs): 157, 163, 190, 187, 183, 173, 184, 189, 193, 192, 177, 173, 168
Question: What is the shape of the height and weight distribution?
Multiple choice:
A. The height and weight distribution exhibit a negative and a positive skew, respectively.
B. Both the height and weight distribution exhibit a positive skew.
C. Both the height and weight distribution exhibit a negative skew.
D. Both the height and weight distribution are symmetric about the mean.
E. The height and weight distribution exhibit a positive and a negative skew, respectively.
To determine the shape of the height and weight distribution, we need to analyze their skewness.
Skewness is a measure of the asymmetry of a probability distribution. A negative skew indicates that the left tail of the distribution is longer or fatter than the right tail, while a positive skew indicates the opposite.
Let's calculate the skewness for both the height and weight distributions:
Height distribution (cm): 178, 163, 174, 186, 154, 167, 167, 181, 159, 165, 177, 191, 158
To calculate skewness, we can use the following formula:
Skewness = (3 * (mean - median)) / standard deviation
Let's calculate the skewness for the height distribution:
1. Calculate the mean:
Mean = (178 + 163 + 174 + 186 + 154 + 167 + 167 + 181 + 159 + 165 + 177 + 191 + 158) / 13 = 171.3077
2. Calculate the median:
The median is the middle value of the sorted data. In this case, there are 13 data points, so the middle value is the 7th value (sorted in ascending order): 167.
3. Calculate the standard deviation:
Using a calculator or spreadsheet software, calculate the standard deviation for the height distribution, which is equal to 11.716
4. Calculate the skewness:
Skewness = (3 * (171.3077 - 167)) / 11.716 = 0.9766
Weight distribution (lbs): 157, 163, 190, 187, 183, 173, 184, 189, 193, 192, 177, 173, 168
1. Calculate the mean:
Mean = (157 + 163 + 190 + 187 + 183 + 173 + 184 + 189 + 193 + 192 + 177 + 173 + 168) / 13 = 179.4615
2. Calculate the median:
The median is the middle value of the sorted data. In this case, there are 13 data points, so the middle value is the 7th value (sorted in ascending order): 183.
3. Calculate the standard deviation:
Using a calculator or spreadsheet software, calculate the standard deviation for the weight distribution, which is equal to 11.608.
4. Calculate the skewness:
Skewness = (3 * (179.4615 - 183)) / 11.608 = -0.5382
Based on the calculations, we find that the skewness of the height distribution is 0.9766, indicating a positive skew. The skewness of the weight distribution is -0.5382, indicating a negative skew.
Therefore, the correct answer is:
E. The height and weight distribution exhibit a positive and a negative skew, respectively.
To determine the shape of a distribution, we need to consider skewness. Skewness measures the asymmetry of a distribution.
To calculate skewness, we can use the following formula:
Skewness = (3 * (Mean – Median)) / Standard Deviation
Let's calculate the skewness for both the height and weight distributions:
For the height distribution:
1. Sort the heights in ascending order: 154, 158, 159, 163, 165, 167, 167, 174, 177, 178, 181, 186, 191.
2. Calculate the mean: Sum of the heights / Total number of heights.
Mean = (154+158+159+163+165+167+167+174+177+178+181+186+191) / 13 = 174.85
3. Find the median: Middle value of the heights.
Median = (165 + 167) / 2 = 166
4. Calculate the standard deviation:
- Calculate the variance: Sum of squares of differences between each height and the mean / Total number of heights.
Variance = ((154-174.85)^2 + (158-174.85)^2 + ... + (191-174.85)^2) / 13 = 83.54
- Take the square root of the variance to get the standard deviation: sqrt(83.54) = 9.14
5. Calculate the skewness:
Skewness = (3 * (174.85 - 166)) / 9.14 ≈ 0.93
For the weight distribution:
1. Sort the weights in ascending order: 157, 163, 168, 173, 173, 177, 183, 184, 187, 189, 190, 192, 193.
2. Calculate the mean: Sum of the weights / Total number of weights.
Mean = (157+163+168+173+173+177+183+184+187+189+190+192+193) / 13 = 178.85
3. Find the median: Middle value of the weights.
Median = (177 + 183) / 2 = 180
4. Calculate the standard deviation:
- Calculate the variance: Sum of squares of differences between each weight and the mean / Total number of weights.
Variance = ((157-178.85)^2 + (163-178.85)^2 + ... + (193-178.85)^2) / 13 = 100.62
- Take the square root of the variance to get the standard deviation: sqrt(100.62) = 10.03
5. Calculate the skewness:
Skewness = (3 * (178.85 - 180)) / 10.03 ≈ -0.35
Based on the calculated skewness values, we can conclude that the shape of the height distribution is positively skewed (skewness ≈ 0.93), while the shape of the weight distribution is negatively skewed (skewness ≈ -0.35).
Therefore, the correct answer is:
E. The height and weight distribution exhibit a positive and a negative skew, respectively.