Compute The Exclusive And Inclusive Ranges For The Following
compute Theexclusiveandinclusiveranges For The Following Itemshigh S
1. Compute the exclusive and inclusive ranges for the following items: high score, low score.
2. Why would you expect more variability on a measure of personality in college freshman grades than on a measure of height?
3. Why does the standard deviation get smaller as the individuals in a group score more similarly on a test?
4. Compute range, unbiased and biased standard deviation and variance on the following numbers: 31, 42, 35, 55, 54, 34, 25, 44, 35.
5. Compute unbiased estimate of standard deviation on the following numbers: 4, 5, 6, 2, 5, 7, 5, 6, 8, 5.
6. Find the range, standard deviation, and variance for the following data sets:
- a. 3, 5, 7, 9
- b. 2, 4, 6, 8
- c. 3.5, 6.2, 9.3, 4.1, 5.5, 7.9
Paper For Above instruction
Understanding the concepts of range, standard deviation, and variance is fundamental in descriptive statistics, providing insights into the spread and variability of data. This paper addresses specific calculations and explanations related to these measures across various data sets, emphasizing the differences between inclusive and exclusive ranges, variability in different contexts, and the mathematical underpinnings of statistical dispersion.
Inclusive and Exclusive Ranges
The inclusive range includes both the highest and lowest scores, calculated as the difference between the maximum and minimum values in a data set. Conversely, the exclusive range often refers to calculating the difference excluding certain extreme values, or sometimes it is synonymous with the simple difference between the highest and lowest scores, depending on context. For the scores, if the high score is 55 and the low score is 25, then:
- Inclusive Range = 55 - 25 = 30
- Exclusive Range = 55 - 25 = 30
Since the problem states 'exclusive and inclusive ranges,' and no further specifics are provided, we usually understand them as the same calculation unless specified otherwise.
Variability in Measures of Personality vs. Height
Personality measures tend to exhibit greater variability in college freshmen compared to physical attributes like height because personality traits are influenced by a complex interplay of genetic, environmental, cultural, and personal experiences. These traits are more susceptible to change and diversification during adolescence and early adulthood. Conversely, height is largely determined by genetics and environmental factors such as nutrition, which tend to result in less variability within a specific age group and population, especially in a relatively homogeneous sample like college freshmen.
Standard Deviation and Group Similarity
The standard deviation reflects how spread out the scores are around the mean. As individuals in a group become more similar in their scores, the amount of deviation from the mean decreases, leading to a smaller standard deviation. This happens because less variation among scores means individual data points are closer to the average, reducing overall variability. Consequently, in groups where members perform similarly, measures of dispersion like standard deviation diminish, indicating less diversity in scores.
Computations for Data Sets
Numbers: 31, 42, 35, 55, 54, 34, 25, 44, 35
Range: 55 - 25 = 30
Mean: (31 + 42 + 35 + 55 + 54 + 34 + 25 + 44 + 35) / 9 ≈ 37.22
Variance (biased):
- Calculate squared deviations:
- (31-37.22)² ≈ 38.66
- (42-37.22)² ≈ 23.31
- (35-37.22)² ≈ 4.89
- (55-37.22)² ≈ 317.19
- (54-37.22)² ≈ 282.89
- (34-37.22)² ≈ 10.37
- (25-37.22)² ≈ 147.58
- (44-37.22)² ≈ 45.19
- (35-37.22)² ≈ 4.89
Sum of squared deviations ≈ 878.87
Biased variance = sum / n = 878.87 / 9 ≈ 97.65
Standard deviation (biased) = √97.65 ≈ 9.88
Unbiased variance (using n-1) = 878.87 / 8 ≈ 109.86
Unbiased standard deviation ≈ √109.86 ≈ 10.48
Numbers: 4, 5, 6, 2, 5, 7, 5, 6, 8, 5
Mean: (4 + 5 + 6 + 2 + 5 + 7 + 5 + 6 + 8 + 5)/10 = 5.3
Calculate squared deviations:
- (4-5.3)² ≈ 1.69
- (5-5.3)² ≈ 0.09
- (6-5.3)² ≈ 0.49
- (2-5.3)² ≈ 10.89
- (5-5.3)² ≈ 0.09
- (7-5.3)² ≈ 2.89
- (5-5.3)² ≈ 0.09
- (6-5.3)² ≈ 0.49
- (8-5.3)² ≈ 7.29
- (5-5.3)² ≈ 0.09
Sum of squared deviations ≈ 23.10
Unbiased variance = 23.10 / (10 - 1) ≈ 2.57
Standard deviation ≈ √2.57 ≈ 1.60
Data Sets and Their Dispersion Measures
Dataset A: 3, 5, 7, 9
Range: 9 - 3 = 6
Mean: (3 + 5 + 7 + 9) / 4 = 6
Squared deviations:
- (3-6)² = 9
- (5-6)² = 1
- (7-6)² = 1
- (9-6)² = 9
Sum: 20
Unbiased variance: 20 / (4 - 1) ≈ 6.67
Standard deviation: √6.67 ≈ 2.58
Dataset B: 2, 4, 6, 8
Range: 8 - 2 = 6
Mean: (2 + 4 + 6 + 8) / 4 = 5
Squared deviations:
- (2-5)² = 9
- (4-5)² = 1
- (6-5)² = 1
- (8-5)² = 9
Sum: 20
Unbiased variance: 20 / 3 ≈ 6.67
Standard deviation: √6.67 ≈ 2.58
Dataset C: 3.5, 6.2, 9.3, 4.1, 5.5, 7.9
Range: 9.3 - 3.5 = 5.8
Mean: (3.5 + 6.2 + 9.3 + 4.1 + 5.5 + 7.9) / 6 ≈ 6.02
Squared deviations:
- (3.5-6.02)² ≈ 6.15
- (6.2-6.02)² ≈ 0.04
- (9.3-6.02)² ≈ 10.77
- (4.1-6.02)² ≈ 3.68
- (5.5-6.02)² ≈ 0.27
- (7.9-6.02)² ≈ 3.56
Sum of squared deviations ≈ 24.47
Unbiased variance: 24.47 / (6 - 1) ≈ 4.89
Standard deviation: √4.89 ≈ 2.21
Conclusion
This analysis demonstrates how measures of dispersion—range, variance, and standard deviation—quantify the variability within different data sets, offering insights into their distribution and spread. Understanding these statistical tools is crucial for interpreting data accurately across various fields.
References
- Everitt, B. S. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press.
- Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for the Behavioral Sciences. Cengage Learning.
- Hays, W. L. (1994). Statistics. Holt, Rinehart and Winston.
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage.
- Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics. W. H. Freeman.
- Witte, R. S., & Witte, J. S. (2017). Statistics. Wiley.
- Rice, J. A. (2007). Mathematical Statistics and Data Analysis. Cengage Learning.
- Ott, L., & Longnecker, M. (2015). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
- Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
- Loftus, G. R. (2014). Observational and Experimental Methods in Behavioral Research. Springer.