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Calculate quartiles, deciles, and percentiles for a dataset of daily sales revenue.
Published on March 23, 2025
Statistics
The solution calculates quartiles, deciles, percentiles, and the coefficient of skewness for a dataset of 21 values. It determines the dataset is negatively skewed.
Question
Answer
Here’s the step-by-step solution to the problem:
a) Calculate the quartiles and deciles for the dataset.
First, we need to 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
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
The number of data points,
-
Quartiles (Q1, Q2, Q3):
- Position of Q1 =
.
Q1 is the average of the 5th and 6th values.
Q1 = - Position of Q2 =
.
Q2 is the 11th value.
Q2 = - Position of Q3 =
.
Q3 is the average of the 16th and 17th values.
Q3 =
- Position of Q1 =
-
Deciles (D1, D5, D9):
- Position of D1 =
.
D1 = 2nd value +(3rd value - 2nd value)
D1 = - Position of D5 =
.
D5 is the 11th value.
D5 = - Position of D9 =
.
D9 = 19th value +(20th value - 19th value)
D9 =
- Position of D1 =
b) Find the 25th, 50th, and 75th percentiles and show that P75 = Q3.
-
Percentiles (P25, P50, P75):
- Position of P25 =
.
P25 = Q1 = - Position of P50 =
.
P50 = Q2 = - Position of P75 =
.
P75 = Q3 =
We can see that P75 = Q3 =. - Position of P25 =
c) Determine the relationship between the measures Me, D5, P50, and Q2.
- Me represents the Median.
From our calculations, we have:
Median (Me) = Q2 = D5 = P50 =.
Therefore, Me = D5 = P50 = Q2.
d) Find the coefficient of skewness using Pearson’s estimate and conclude about the shape of the distribution.
-
Mean (
):
-
Median (Me):
Me = Q2 = -
Standard Deviation (s):
First, calculate the variance ():
(using sample standard deviation as it’s a dataset)
Using the sum of squares calculated in the thought process:
-
Pearson’s Coefficient of Skewness (Sk):
Sk =Since the coefficient of skewness is negative (approximately -0.358), the distribution is negatively skewed or skewed to the left. This indicates that the tail of the distribution is longer on the left side, and the mean is less than the median.
Final Answer: The final answer is