Given The Data Set: 5, 4, 6, 14, 2, 2, 11, 4, 5, 8, 3, 1, 12
Given The Data Set A 5 4 6 14 2 2 11 4 5 8 3 1 12 15 13
Given the data set A = {5, 4, 6, 14, 2, 2, 11, 4, 5, 8, 3, 1, 12, 15, 13}, which is the data of a sample taken from a larger population.
Calculate the arithmetic mean. Show all work!
Find the median. Show all work!
Find the mode. Show all work!
Calculate the range. Show all work!
Calculate the interquartile range. Show all work!
Calculate the mean deviation. Show all work!
Calculate the variance. Show all work!
Calculate the standard deviation. Show all work!
Paper For Above instruction
Introduction
Descriptive statistics are essential tools in understanding the distribution and characteristics of data sets. They provide metrics that summarize the data, including measures of central tendency—mean, median, and mode—and measures of dispersion such as range, interquartile range, mean deviation, variance, and standard deviation. This paper presents detailed calculations for each of these statistical measures based on the provided data set A = {5, 4, 6, 14, 2, 2, 11, 4, 5, 8, 3, 1, 12, 15, 13}.
Calculating the Arithmetic Mean
The arithmetic mean is computed by summing all data points and dividing by the number of observations.
Sum of data points:
5 + 4 + 6 + 14 + 2 + 2 + 11 + 4 + 5 + 8 + 3 + 1 + 12 + 15 + 13 = 105
Number of observations:
15
Mean:
Mean = Total Sum / Number of observations = 105 / 15 = 7
Calculating the Median
The median is the middle value when the data is ordered.
Ordered data set:
1, 2, 2, 3, 4, 4, 5, 5, 6, 8, 11, 12, 13, 14, 15
Since there are 15 data points (odd number), the median is the 8th value:
Median = 5
Calculating the Mode
The mode is the value(s) that appear most frequently.
Frequency counts:
- 1: 1
- 2: 2
- 3: 1
- 4: 2
- 5: 2
- 6: 1
- 8: 1
- 11: 1
- 12: 1
- 13: 1
- 14: 1
- 15: 1
Multiple values (2, 4, 5) appear twice; hence, the data set is multimodal with modes at 2, 4, and 5.
Calculating the Range
Range is the difference between the maximum and minimum values.
Maximum value = 15
Minimum value = 1
Range = 15 - 1 = 14
Calculating the Interquartile Range (IQR)
First, identify Q1 (lower quartile) and Q3 (upper quartile).
- Lower half (first 7 values): 1, 2, 2, 3, 4, 4, 5
- Upper half (last 8 values): 5, 6, 8, 11, 12, 13, 14, 15
Q1 (median of lower half):
Values: 1, 2, 2, 3, 4, 4, 5
Median of lower half (positions 4): 3
Q3 (median of upper half):
Values: 5, 6, 8, 11, 12, 13, 14, 15
Median of upper half (positions 4 and 5): (11 + 12) / 2 = 11.5
IQR = Q3 - Q1 = 11.5 - 3 = 8.5
Calculating the Mean Deviation
Mean deviation is the average of absolute deviations from the mean.
|Data point - Mean|:
|5 - 7| = 2
|4 - 7| = 3
|6 - 7| = 1
|14 - 7| = 7
|2 - 7| = 5
|2 - 7| = 5
|11 - 7| = 4
|4 - 7| = 3
|5 - 7| = 2
|8 - 7| = 1
|3 - 7| = 4
|1 - 7| = 6
|12 - 7| = 5
|15 - 7|= 8
|13 - 7|= 6
Sum of absolute deviations:
2 + 3 + 1 + 7 + 5 + 5 + 4 + 3 + 2 + 1 + 4 + 6 + 5 + 8 + 6 = 72
Mean deviation = Total absolute deviation / Number of observations = 72 / 15 = 4.8
Calculating the Variance
Variance measures the average squared deviation from the mean.
Squared deviations:
(5 - 7)^2 = 4
(4 - 7)^2 = 9
(6 - 7)^2 = 1
(14 - 7)^2 = 49
(2 - 7)^2 = 25
(2 - 7)^2 = 25
(11 - 7)^2 = 16
(4 - 7)^2 = 9
(5 - 7)^2= 4
(8 - 7)^2= 1
(3 - 7)^2= 16
(1 - 7)^2= 36
(12 - 7)^2= 25
(15 -7)^2= 64
(13 -7)^2= 36
Sum of squared deviations:
4 + 9 + 1 + 49 + 25 + 25 + 16 + 9 + 4 + 1 + 16 +36 +25 +64 +36 = 340
Sample variance (dividing by n-1):
Variance = 340 / (15 - 1) = 340 / 14 ≈ 24.29
Calculating the Standard Deviation
Standard deviation is the square root of the variance:
Standard deviation = √24.29 ≈ 4.93
Conclusion
The data set displays a mean of 7, a median of 5, and multiple modes at 2, 4, and 5. The data range is 14, with an interquartile range of 8.5, indicating moderate dispersion. The mean absolute deviation is 4.8, and the variance and standard deviation are approximately 24.29 and 4.93, respectively, reflecting the spread of the data points around the mean.
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