Determining the Spread of a Distribution Variance & Standard Deviation

Transcription

1 Determining the Spread of a Distribution Variance & Standard Deviation 1.3 Cathy Poliak, Ph.D. cathy@math.uh.edu Department of Mathematics University of Houston Lecture 3 Lecture 3 1 / 32

2 Outline 1 Describing Quantitative Variables 2 Understanding Standard Deviation 3 Calculating The Standard Deviation 4 Coefficient of Variation Lecture 3 2 / 32

3 Describing distributions of quantitative variables The distribution of a variable tells us what values it takes and how often it takes these values. There are four main characteristics to describe a distribution: 1. Shape 2. Center 3. Spread 4. Outliers Lecture 3 3 / 32

4 Describing distributions An initial view of the distribution and the characteristics can be shown through the graphs. Then we use numerical descriptions to get a better understanding of the distributions characteristics. Lecture 3 4 / 32

5 Describing Center with Numbers Mean: Sum all of the numbers then divide by number of values in the data set. Median: The center value of ordered data. Mode: The value that has the highest frequency in the data. Lecture 3 5 / 32

6 Types of Measurements for the Spread Range Percentiles Quartiles IQR; Interquartile range Variance Standard deviation Coefficient of Variation Lecture 3 6 / 32

7 Measuring Spread: The Standard Deviation Measures spread by looking at how far the observations are from their mean. Most common numerical description for the spread of a distribution. A larger standard deviation implies that the values have a wider spread from the mean. Denoted s when used with a sample. This is the one we calculate from a list of values. Denoted σ when used with a population. This is the "idealized" standard deviation. The standard deviation has the same units of measurements as the original observations. Lecture 3 7 / 32

8 Definition of the Standard Deviation The standard deviation is the average distance each observation is from the mean. Using this list of values from a sample: 3, 3, 9, 15, 15 The mean is 9. By definition, the average distance each of these values are from the mean is 6. So the standard deviation is 6. Lecture 3 8 / 32

9 Values of the Standard Deviation The standard deviation is a value that is greater than or equal to zero. It is equal to zero only when all of the observations have the same value. By the definition of standard deviation determine s for the following list of values. 2, 2, 2, 2 : standard deviation = 0 125, 125, 125, 125, 125: standard deviation = 0 Lecture 3 9 / 32

10 Adding or Subtracting a Value to the Observations Adding or subtracting the same value to all the original observations does not change the standard deviation of the list. Using this list of values: 3, 3, 9, 15, 15 mean = 9, standard deviation = 6. If we add 4 to all the values: 7, 7, 13, 19, 19 mean = 13, standard deviation = 6 Lecture 3 10 / 32

11 Multiplying or Dividing a Value to the Observations Multiplying or dividing the same value to all the original observations will change the standard deviation by that factor. Using this list of values: 3, 3, 9, 15, 15: mean = 9, standard deviation = 6. If we double all the values: 6, 6, 18, 30, 30 mean = 18, standard deviation = 12 athy Poliak, Ph.D. cathy@math.uh.edu (Department of Mathematics 1.3 University of Houston ) Lecture 3 11 / 32

12 Population Variance and Standard Deviation If N is the number of values in a population with mean mu, and x i represents each individual in the population, the the population variance is found by: σ 2 = N i=1 (x i µ) 2 N and the population standard deviation is the square root, σ = σ 2. Lecture 3 12 / 32

13 Sample Variance and Standard Deviation Most of the time we are working with a sample instead of a population. So the sample variance is found by: s 2 = n i=1 (x i x) 2 n 1 and the sample standard deviation is the square root, s = s 2. Where n is the number of observations (samples), x i is the value for the i th observation and x is the sample mean. Lecture 3 13 / 32

14 Calculating the Standard Deviation By Hand When calculating by hand we will calculate s. 1. Find the mean of the observations x. 2. Calculate the difference between the observations and the mean for each observation x i x. This is called the deviations of the observations. 3. Square the deviations for each observation (x i x) Add up the squared deviations together n i=1 (x i x) Divide the sum of the squared deviations by one less than the number of observations n 1. This is the variance s 2 = 1 n 1 n (x i x) 2 i=1 athy Poliak, Ph.D. cathy@math.uh.edu (Department of Mathematics 1.3 University of Houston ) Lecture 3 14 / 32

15 Step 6: Standard Deviation 6. Find the square root of the variance. This is the standard deviation s = 1 n (x i x) n 1 2 i=1 Lecture 3 15 / 32

16 Example: Section A Determine the sample standard deviation of the test scores for Section A. Section A Scores (X i ) Lecture 3 16 / 32

17 Step 1: Calculate the Mean The sample mean is x = Lecture 3 17 / 32

18 Step 2: Calculate Deviations For All Values Variable Deviations Deviations Squared Score (X i ) X i X (X i X) = = = = = = = = = = 5.5 sum Lecture 3 19 / 32

19 Step 3: Calculate Squared Deviations Variable Deviations Deviations Squared Score (X i ) X i X (X i X) = 6.5 ( 6.5) 2 = = 5.5 ( 5.5) 2 = = 4.5 ( 4.5) 2 = = 3.5 ( 3.5) 2 = = 0.5 ( 0.5) 2 = = = = = = = = = = = sum athy Poliak, Ph.D. cathy@math.uh.edu (Department of Mathematics 1.3 University of Houston ) Lecture 3 20 / 32

20 Step 4: Calculate the Sum of the Squared Deviations Variable Deviations Deviations Squared Score(X i ) X i X (X i X) = 6.5 ( 6.5) 2 = = 5.5 ( 5.5) 2 = = 4.5 ( 4.5) 2 = = 3.5 ( 3.5) 2 = = 0.5 ( 0.5) 2 = = = = = = = = = = = sum n i=1 (X i X) 2 = athy Poliak, Ph.D. cathy@math.uh.edu (Department of Mathematics 1.3 University of Houston ) Lecture 3 21 / 32

21 Step 5: Calculate the Variance variance = s 2 = 1 n 1 n (x i x) 2 i=1 = = Lecture 3 22 / 32

22 Step 6: Take the Square Root of the Variance standard deviation = s = 1 n 1 = = 4.77 n (x i x) 2 i=1 Lecture 3 23 / 32

23 Sample Standard Deviation of Section A test scores Sample standard deviation is s = This implies that from the sample of the 10 students from section A the tests scores has a spread, on average, of 4.77 points from the mean of points. athy Poliak, Ph.D. cathy@math.uh.edu (Department of Mathematics 1.3 University of Houston ) Lecture 3 24 / 32

24 Example A statistics teacher wants to decide whether or not to curve an exam. From her class of 300 students, she chose a sample of 10 students and their grade were: 72, 88, 85, 81, 60, 54, 70, 72, 63, 43 Determine the sample mean. What is the variance? What is the standard deviation? athy Poliak, Ph.D. cathy@math.uh.edu (Department of Mathematics 1.3 University of Houston ) Lecture 3 25 / 32

25 Add 10 Suppose the statistics instructor decides to curve the grade by adding 10 points to each score. What is the new mean, variance and standard deviation? Lecture 3 26 / 32

26 Multiply by 2 For the following dataset the mean is x = 4.5, the variance is s 2 = 3.5 and the standard deviation is s = , 6, 2, 7, 4, 5 Now, multiply each value by 2. What is the new variance and the new standard deviation? Lecture 3 27 / 32

27 Calculating Standard Deviation For larger data sets use a calculator or computer software. Each calculator is different if you cannot determine how to compute standard deviation from your calculator ask your instructor. For this course we will be using R as the software. The function for the sample standard deviation in R is sd(data name$variable name). athy Poliak, Ph.D. cathy@math.uh.edu (Department of Mathematics 1.3 University of Houston ) Lecture 3 28 / 32

28 Coefficient of Variation This is to compare the variation between two groups. The coefficient of variation (cv) is the ratio of the standard deviation to the mean. cv = sd mean A smaller ratio will indicate less variation in the data. Lecture 3 29 / 32

29 CV of test scores Section A Section B Sample Size Sample Mean Sample Standard Deviation CV 71.5 = = Lecture 3 30 / 32

30 CV Example The following statistics were collected on two different groups of stock prices: Portfolio A Portfolio B Sample size Sample mean $52.65 $49.80 Sample standard deviation $6.50 $2.95 What can be said about the variability of each portfolio? Lecture 3 31 / 32