1 / 27

MEASURES OF VARIABILITY

MEASURES OF VARIABILITY. Variance Population variance Sample variance Standard Deviation Population standard deviation Sample standard deviation Coefficient of Variation (CV) Sample CV Population CV. MEASURES OF VARIABILITY POPULATION VARIANCE.

presley
Download Presentation

MEASURES OF VARIABILITY

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. MEASURES OF VARIABILITY • Variance • Population variance • Sample variance • Standard Deviation • Population standard deviation • Sample standard deviation • Coefficient of Variation (CV) • Sample CV • Population CV

  2. MEASURES OF VARIABILITYPOPULATION VARIANCE • The population variance is the mean squared deviation from the population mean: • Where 2stands for the population variance •  is the population mean • N is the total number of values in the population • is the value of the i-th observation. • represents a summation

  3. MEASURES OF VARIABILITYSAMPLE VARIANCE • The sample variance is defined as follows: • Where s2stands for the sample variance • is the sample mean • n is the total number of values in the sample • is the value of the i-th observation. • represents a summation

  4. MEASURES OF VARIABILITYSAMPLE VARIANCE • A sample of monthly advertising expenses (in 000$) is taken. The data for five months are as follows: 2.5, 1.3, 1.4, 1.0 and 2.0. Compute the sample variance.

  5. MEASURES OF VARIABILITYSAMPLE VARIANCE • Notice that the sample variance is defined as the sum of the squared deviations divided by n-1. • Sample variance is computed to estimate the population variance. • An unbiased estimate of the population variance may be obtained by defining the sample variance as the sum of the squared deviations divided by n-1 rather than by n. • Defining sample variance as the mean squared deviation from the sample mean tends to underestimate the population variance.

  6. MEASURES OF VARIABILITYSAMPLE VARIANCE • A shortcut formula for the sample variance: • Where s2is the sample variance • n is the total number of values in the sample • is the value of the i-th observation. • represents a summation

  7. MEASURES OF VARIABILITYSAMPLE VARIANCE • A sample of monthly sales expenses (in 000 units) is taken. The data for five months are as follows: 264, 116, 165, 101 and 209. Compute the sample variance using the short-cut formula.

  8. MEASURES OF VARIABILITYSAMPLE VARIANCE • The shortcut formula for the sample variance: • If you have the sum of the measurements already computed, the above formula is a shortcut because you need only to compute the sum of the squares,

  9. MEASURES OF VARIABILITY POPULATION/SAMPLE STANDARD DEVIATION • The standard deviation is the positive square root of the variance: Population standard deviation: Sample standard deviation: • Compute the standard deviations of advertising and sales.

  10. MEASURES OF VARIABILITY POPULATION/SAMPLE STANDARD DEVIATION • Compute the sample standard deviation of advertising data: 2.5, 1.3, 1.4, 1.0 and 2.0 • Compute the sample standard deviation of sales data: 264, 116, 165, 101 and 209

  11. MEASURES OF VARIABILITY POPULATION/SAMPLE CV • The coefficient of variation is the standard deviation divided by the means Population coefficient of variation: Sample coefficient of variation:

  12. MEASURES OF VARIABILITY POPULATION/SAMPLE CV • Compute the sample coefficient of variation of advertising data: 2.5, 1.3, 1.4, 1.0 and 2.0 • Compute the sample coefficient of variation of sales data: 264, 116, 165, 101 and 209

  13. MEASURES OF ASSOCIATION • Scatter diagram plot provides a graphical description of positive/negative, linear/non-linear relationship • Some numerical description of the positive/negative, linear/non-linear relationship are obtained by: • Covariance • Population covariance • Sample covariance • Coefficient of correlation • Population coefficient of correlation • Sample coefficient of correlation

  14. Sales Advertising Month (000 units) (000 $) 1 264 2.5 2 116 1.3 3 165 1.4 4 101 1.0 5 209 2.0 MEASURES OF ASSOCIATION: EXAMPLE • A sample of monthly advertising and sales data are collected and shown below: • How is the relationship between sales and advertising? Is the relationship linear/non-linear, positive/negative, etc.

  15. POPULATION COVARIANCE • The population covariance is mean of products of deviations from the population mean: • Where COV(X,Y) is the population covariance • x,y are the population means of X and Y respectively • N is the total number of values in the population • are the values of the i-th observations of X and Y respectively. • represents a summation

  16. SAMPLE COVARIANCE • The sample covariance is mean of products of deviations from the sample mean: • Where cov(X,Y) is the sample covariance • are the sample means of X and Y respectively • n is the total number of values in the population • are the values of the i-th observations of X and Y respectively. • represents a summation

  17. SAMPLE COVARIANCE

  18. POPULATION/SAMPLE COVARIANCE • If two variables increase/decrease together, covariance is a large positive number and the relationship is called positive. • If the relationship is such that when one variable increases, the other decreases and vice versa, then covariance is a large negative number and the relationship is called negative. • If two variables are unrelated, the covariance may be a small number. • How large is large? How small is small?

  19. POPULATION/SAMPLE COVARIANCE • How large is large? How small is small? A drawback of covariance is that it is usually difficult to provide any guideline how large covariance shows a strong relationship and how small covariance shows no relationship. • Coefficient of correlation can overcome this drawback to a certain extent.

  20. POPULATION COEFFICIENT OF CORRELATION • The population coefficient of correlation is the population covariance divided by the population standard deviations of X and Y: • Where  is the population coefficient of correlation • COV(X,Y) is the population covariance • x,y are the population means of X and Y respectively

  21. SAMPLE COEFFICIENT OF CORRELATION • The sample coefficient of correlation is the sample covariance divided by the sample standard deviations of X and Y: • Where r is the sample coefficient of correlation • cov(X,Y) is the sample covariance • sx,sy are the sample means of X and Y respectively

  22. SAMPLE COEFFICIENT OF CORRELATION

  23. POPULATION/SAMPLE COEFFICIENT OF CORRELATION • The coefficient of correlation is always between -1 and +1. • Values near -1 or +1 show strong relationship • Values near 0 show no relationship’ • Values near 1 show strong positive linear relationship • Values near -1 show strong negative linear relationship

  24. EXAMPLE • Salary and expenses for cultural activities, and sports related activities are collected from 100 households. Data of only 5 households shown below: How are the relationships (linear/non-linear, positive/negative) between (i) salary and culture, (ii) salary and sports, and (iii) sports and culture?

  25. cov = 1094787, r = 0.5065 (positive, linear)

  26. cov = -33608, r = -0.5201 (negative, linear)

  27. cov = -219026, r = -0.08122 (no linear relationship)

More Related