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Calculate quartiles, percentiles, and coefficient of skewness for a dataset of daily sales revenue.
Published on March 16, 2025
Statistics
The solution calculates quartiles, deciles, percentiles, and the coefficient of skewness for a dataset of 21 values. It demonstrates the relationships between these measures and concludes that the distribution is slightly negatively skewed.
Question
Answer
Here’s the step-by-step solution to the problem:
a) Calculate the quartiles and deciles (D1, D5, D9) for the dataset.
First, sort the dataset in ascending order:
12.47, 15.93, 24.78, 25.37, 31.76, 37.62, 39.28, 47.58, 49.05, 53.89, 56.72, 58.34, 62.94, 66.81, 68.29, 71.05, 72.15, 79.23, 83.47, 84.61, 90.56
Number of data points,
12.47, 15.93, 24.78, 25.37, 31.76, 37.62, 39.28, 47.58, 49.05, 53.89, 56.72, 58.34, 62.94, 66.81, 68.29, 71.05, 72.15, 79.23, 83.47, 84.61, 90.56
Number of data points,
Quartiles:
-
First Quartile (Q1): Position =
.
Q1 is the average of the 5th and 6th values.
Q1 = -
Second Quartile (Q2): Position =
.
Q2 is the 11th value.
Q2 = -
Third Quartile (Q3): Position =
.
Q3 is the average of the 16th and 17th values.
Q3 =
Deciles:
-
First Decile (D1): Position =
.
D1 is the value at position 2 + 0.2 * (value at position 3 - value at position 2).
D1 = -
Fifth Decile (D5): Position =
.
D5 is the 11th value.
D5 = -
Ninth Decile (D9): Position =
.
D9 is the value at position 19 + 0.8 * (value at position 20 - value at position 19).
D9 =
b) Find the 25th percentile (P25), 50th percentile (P50), and 75th percentile (P75) and show that P75 = Q3.
-
25th Percentile (P25): Position =
.
P25 is the average of the 5th and 6th values.
P25 = -
50th Percentile (P50): Position =
.
P50 is the 11th value.
P50 = -
75th Percentile (P75): Position =
.
P75 is the average of the 16th and 17th values.
P75 =
We observe that P75 = 71.60 and Q3 = 71.60. Therefore, P75 = Q3.
c) Determine the relationship between the measures Me, D5, P50, and Q2.
- Median (Me): For a sorted dataset with
(odd), the median is the value, which is . - Fifth Decile (D5): D5 =
(calculated in part a). - 50th Percentile (P50): P50 =
(calculated in part b). - Second Quartile (Q2): Q2 =
(calculated in part a).
Thus, Me = D5 = P50 = Q2 =
d) Find the coefficient of skewness using Pearson’s estimate. What is your conclusion regarding the shape of the distribution?
Pearson’s coefficient of skewness is given by:
First, calculate the Mean of the dataset:
We already know the Median (Q2, D5, P50) =
Next, calculate the Standard Deviation. First, calculate the variance (using sample standard deviation formula, dividing by
Now, calculate Pearson’s coefficient of skewness:
Conclusion regarding the shape of the distribution:
Since the coefficient of skewness is approximately
Since the coefficient of skewness is approximately
Final Answer: The final answer is