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Questions and Answers
What does a measure of dispersion indicate about a data set?
What does a measure of dispersion indicate about a data set?
- The frequency distribution of the data points.
- The central tendency of the data points.
- The variability structure of the data points. (correct)
- The correlation between the data points.
Which measure of absolute dispersion is defined as the difference between the largest and the smallest values?
Which measure of absolute dispersion is defined as the difference between the largest and the smallest values?
- Standard Deviation
- Range (correct)
- Variance
- Mean Deviation
When do measures of dispersion take on the value zero?
When do measures of dispersion take on the value zero?
- When all values are different.
- When the largest value is equal to the smallest value. (correct)
- When calculations involve negative numbers.
- When the data set contains no values.
What is indicated by a large dispersion in a data set?
What is indicated by a large dispersion in a data set?
What type of dispersion measures are used to compare the dispersion between two or more groups?
What type of dispersion measures are used to compare the dispersion between two or more groups?
In a data set with values of -5, 5, 5, and 15, what is the range?
In a data set with values of -5, 5, 5, and 15, what is the range?
Which of the following statements about measures of dispersion is true?
Which of the following statements about measures of dispersion is true?
What is the formula for calculating the range of a data set?
What is the formula for calculating the range of a data set?
Which value is NOT part of the calculation for the Inter-Quartile Range (I.Q.R)?
Which value is NOT part of the calculation for the Inter-Quartile Range (I.Q.R)?
How does the I.Q.R. compare to the range in terms of sensitivity to extreme values?
How does the I.Q.R. compare to the range in terms of sensitivity to extreme values?
What is the standard deviation derived from, according to the content?
What is the standard deviation derived from, according to the content?
Which of the following statements about the I.Q.R. is false?
Which of the following statements about the I.Q.R. is false?
What does a higher standard deviation indicate about a data set?
What does a higher standard deviation indicate about a data set?
Which statistic is calculated from all data points in a population?
Which statistic is calculated from all data points in a population?
In the calculation of the range, which of the following is essential?
In the calculation of the range, which of the following is essential?
What is the symbol for the standard deviation calculated from sample data?
What is the symbol for the standard deviation calculated from sample data?
What is the denominator used when calculating the sample variance?
What is the denominator used when calculating the sample variance?
What is the primary unit in which variance is measured?
What is the primary unit in which variance is measured?
Using the values 4, 9, and 5, what is the mean ($ar{x}$) calculated from these numbers?
Using the values 4, 9, and 5, what is the mean ($ar{x}$) calculated from these numbers?
What is the value of the sample variance ($S^2$) calculated from the data 4, 9, and 5?
What is the value of the sample variance ($S^2$) calculated from the data 4, 9, and 5?
What final value is expected for the sample standard deviation ($S$) from the calculations provided?
What final value is expected for the sample standard deviation ($S$) from the calculations provided?
What is the first step to find descriptive statistics for a given data set?
What is the first step to find descriptive statistics for a given data set?
Which of the following is true regarding sample variance and standard deviation?
Which of the following is true regarding sample variance and standard deviation?
When calculating the value for 𝑺 from the given data, what operation is performed on the variance?
When calculating the value for 𝑺 from the given data, what operation is performed on the variance?
Which of the following measures is sensitive to extreme values?
Which of the following measures is sensitive to extreme values?
If the variance of a variable x is 25, what is the standard deviation of y if y = 9 - 2x?
If the variance of a variable x is 25, what is the standard deviation of y if y = 9 - 2x?
What does the range of a constant data set (e.g. 8, 8, 8) equal?
What does the range of a constant data set (e.g. 8, 8, 8) equal?
How do you calculate the coefficient of variation (CV)?
How do you calculate the coefficient of variation (CV)?
When comparing the dispersion of two groups, why are coefficients of variation preferred?
When comparing the dispersion of two groups, why are coefficients of variation preferred?
If the variance of variable x is 10, what would be the variance of y if y = 5 - 2x?
If the variance of variable x is 10, what would be the variance of y if y = 5 - 2x?
Which country has less inequality in salary distribution with an annual salary of $20,000 (SD = $2000) and $4,000 (SD = $1000)?
Which country has less inequality in salary distribution with an annual salary of $20,000 (SD = $2000) and $4,000 (SD = $1000)?
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Study Notes
Measures of Dispersion
- Measures of dispersion describe how spread out data is around the center
- Dispersion is high when data points are widely scattered
- Dispersion is low when data points are close together
- Absolute dispersion measures describe the variability of one group
- Relative dispersion or coefficients of variation are used to compare the dispersion of two or more groups
- All measures of dispersion are non-negative
- A measure of dispersion is 0 only if all data points are the same value
Range
- Range is the simplest measure of dispersion
- Range is the difference between the largest and smallest values in a data set
- The range is sensitive to extreme values or outliers
Interquartile Range
- Interquartile Range (IQR) measures the range of the middle 50% of the data
- The IQR is calculated as IQR = Q3 - Q1 where Q1 and Q3 are the first and third quartiles
- IQR is insensitive to extreme values or outliers
Standard Deviation and Variance
- Standard Deviation is the most common measure of dispersion
- Standard Deviation tells how closely data points are clustered around the mean
- Variance is the square of the standard deviation
- Population variance is denoted as (σ²) and sample variance is denoted as (S²)
- The standard deviation formulas:
- Population standard deviation: σ = √σ²
- Sample standard deviation: S = √S²
- Sample variance: S² = Σ(x - x̅)² / (n - 1)
Coefficient of Variation (CV)
- CV is used to compare the dispersion of two or more groups
- CV is a unitless measure of dispersion
- It is calculated as the ratio of the measure of absolute dispersion to the corresponding measure of central tendency
- The formula for CV can be written as follows:
- σ / |μ| x 100
- S / |X̅| x 100
- IQR / |Q2| x 100
Properties of Measures of Dispersion
- If all data points in a set are the same, the mean, median, and mode are equal, and all measures of dispersion are equal to zero
- If a linear transformation is applied to a data set, the range, IQR, and Standard Deviation are affected in a predictable way
- If y = a + b.x, then:
- Range(y) = |b|.Range(x)
- IQR(y) = |b|.IQR(x)
- Sd(y) = |b|.Sd(x)
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