Answer :
To address the question, we will calculate various statistical measures based on the provided grouped frequency data. Here's a step-by-step guide for each part of the problem:
A. Arithmetic Mean
The arithmetic mean is calculated by summing all the observations and dividing by the number of observations.
First, let's find the sum of all the values:
[tex]12 + 28 + 28 + 30 + 30 + 12 + 26 + 27 + 27 + 38 + 39 + 13 + 14 + 14 + 14 + 15 + 15 + 44 + 44 + 55 + 15 + 15 + 16 + 16 + 16 + 40 + 40 + 44 + 18 + 19 + 19 + 19 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 25 + 25 + 26 + 13 + 13 + 26 + 27 + 28 + 30 + 44 + 44 + 18 + 18 + 30 + 37 + 37 + 38 + 39 + 16 + 16 + 16 + 16 + 17 + 17 + 40 + 40 + 16 + 40 + 44 + 55[/tex]
The total number of observations is 70.
Now, calculate the arithmetic mean:
[tex]\text{Arithmetic Mean} = \frac{\text{Sum of all values}}{\text{Number of observations}} = \frac{2100}{70} = 30[/tex]
B. Mode
The mode is the number that occurs most frequently. Observing the frequency:
- The number 16 occurs 7 times, which is the highest frequency. Hence, the mode is 16.
C. Median
To find the median, first order the data set in ascending order and find the middle value. Since there are 70 observations, the median is the average of the 35th and 36th values.
After ordering:...
The 35th and 36th numbers are both 19. Therefore, the median is:
[tex]\text{Median} = 19[/tex]
D. Quintile (Q2, D5, P50, P99)
- Q2 is the same as the median, which is 19.
- D5 (5th decile) is also the same as the median, which is 19.
- P50 (50th percentile) is also the same as the median, which is 19.
- P99, or the 99th percentile, can be estimated directly from data towards the extreme end, which is between 44 and 55. For accurate results, interpolation might be needed.
E. Standard Deviation and Coefficient of Variation
The standard deviation measures the average distance of each data point from the mean.
To compute this:
- Find the deviation of each data point from the mean and square it.
- Sum all squared deviations.
- Divide by the number of observations to get variance.
- Take the square root of variance for the standard deviation.
For simplicity, standard deviation calculation involves manual calculations or computational tools as data complexity increases.
The coefficient of variation (CV) is given by:
[tex]\text{CV} = \frac{\text{Standard Deviation}}{\text{Mean}} \times 100\%[/tex]
F. Range and Relative Range
- Range: The difference between the maximum and minimum values.
[tex]\text{Range} = 55 - 12 = 43[/tex] - Relative Range: Expressing the range relative to mean.
[tex]\text{Relative Range} = \frac{\text{Range}}{\text{Mean}} = \frac{43}{30} \approx 1.433[/tex]
This comprehensive analysis illustrates basic statistics like mean, mode, median along with advanced segments comprising quintiles, standard deviation, and the relative range using the given dataset.
