Statistics My name: Huiyuan Liu---刘慧媛 My email: lhyshfc@126.com My address: Room 307 No.1 Teaching Building

Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.

Course Content 1 Data and Statistics 2 Tabular and Graphical Presentation 3 Numerical Measures 7 Sampling and Sampling Distributions 8 Interval Estimation 9 Hypothesis Tests: one-Sample Tests 14 Simple Linear Regression

Learning Objectives In this chapter, you learn: To describe the properties of central tendency, variation, and shape in numerical data To construct and interpret a boxplot To calculate the covariance and the coefficient of correlation Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.

Summary Definitions The central tendency is the extent to which all the data values group around a typical or central value. The variation is the amount of dispersion, or scattering, of values The shape is the pattern of the distribution of values from the lowest value to the highest value.

Measures of Central Tendency: The Mean The arithmetic mean (often just called “mean”) is the most common measure of central tendency For sample mean: Pronounced x-bar The ith value Sample size Observed values

Numerical Descriptive Measures for a Population: The mean µ For population mean Where μ = population mean N = population size Xi = ith value of the variable X

Measures of Central Tendency: The Mean (continued) The most common measure of central tendency Mean = sum of values divided by the number of values Affected by extreme values (outliers) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Mean = 3 Mean = 4

Measures of Central Tendency: The Median In an ordered array, the median is the “middle” number (50% above, 50% below) Not affected by extreme values 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Median = 3 Median = 3

Measures of Central Tendency: Locating the Median The location of the median when the values are in numerical order (smallest to largest): If the number of values is odd, the median is the middle number If the number of values is even, the median is the average of the two middle numbers Note that is not the value of the median, only the position of the median in the ranked data

Measures of Central Tendency: The Mode Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may be no mode There may be several modes 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 No Mode Mode = 9

Measures of Central Tendency: Review Example House Prices: $2,000,000 $500,000 $300,000 $100,000 $100,000 Sum $3,000,000 Mean: ($3,000,000/5) = $600,000 Median: middle value of ranked data = $300,000 Mode: most frequent value = $100,000

Measures of Central Tendency: Which Measure to Choose? The mean is generally used, unless extreme values (outliers) exist. The median is often used, since the median is not sensitive to extreme values. For example, median home prices may be reported for a region; it is less sensitive to outliers. In some situations it makes sense to report both the mean and the median.

Measures of Central Tendency: Summary Arithmetic Mean Median Mode Most frequently observed value Middle value in the ordered array

Coefficient of Variation Measures of Variation Variation Standard Deviation Coefficient of Variation Range Variance Measures of variation give information on the spread or variability or dispersion of the data values. Same center, different variation

Measures of Variation: The Range Simplest measure of variation Difference between the largest and the smallest values: Range = Xlargest – Xsmallest Example: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Range = 13 - 1 = 12 Indicate the largest difference between any two data

Measures of Variation: Why The Range Can Be Misleading Ignores the way in which data are distributed Sensitive to outliers 7 8 9 10 11 12 7 8 9 10 11 12 Range = 12 - 7 = 5 Range = 12 - 7 = 5 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5 Range = 5 - 1 = 4 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120 Range = 120 - 1 = 119

Measures of Variation: The Variance Average (approximately) of squared deviations of values from the mean Sample variance: Where = arithmetic mean n = sample size Xi = ith value of the variable X

Measures of Variation: The Standard Deviation Most commonly used measure of variation Shows variation about the mean Is the square root of the variance Has the same units as the original data Sample standard deviation:

Measures of Variation: The Standard Deviation Steps for Computing Standard Deviation 1. Compute the difference between each value and the mean. 2. Square each difference. 3. Add the squared differences. 4. Divide this total by n-1 to get the sample variance. 5. Take the square root of the sample variance to get the sample standard deviation.

Measures of Variation: Sample Standard Deviation: Calculation Example Sample Data (Xi) : 10 12 14 15 17 18 18 24 n = 8 Mean = X = 16 A measure of the “average” scatter around the mean

Measures of Variation: Comparing Standard Deviations Data A Mean = 15.5 11 12 13 14 15 16 17 18 19 20 21 S = 3.338 Data B Mean = 15.5 S = 0.926 11 12 13 14 15 16 17 18 19 20 21 Data C Mean = 15.5 S = 4.570 11 12 13 14 15 16 17 18 19 20 21 The more data are dispersed, the larger the standard deviation.

Measures of Variation: Comparing Standard Deviations Data are concentrated, Smaller standard deviation Data are dispersed, Larger standard deviation

Measures of Population Variation: The Variance σ2 Average of squared deviations of values from the mean Population variance: Where μ = population mean N = population size Xi = ith value of the variable X

Measures of Population Variation: The Standard Deviation σ Most commonly used measure of variation Shows variation about the mean Is the square root of the population variance Has the same units as the original data Population standard deviation:

Measures of Variation: Summary Characteristics The more the data are spread out, the greater the range, variance, and standard deviation. The more the data are concentrated, the smaller the range, variance, and standard deviation. If the values are all the same (no variation), all these measures will be zero. None of these measures are ever negative.

Sample statistics versus population parameters Measure Population Parameter Sample Statistic Mean Variance Standard Deviation

Measures of Variation: The Coefficient of Variation Measures relative variation Always in percentage (%) Shows variation relative to mean Can be used to compare the variability of two or more sets of data measured in different units

Measures of Variation: Comparing Coefficients of Variation Stock A: Average price last year = $50 Standard deviation = $5 Stock B: Average price last year = $100 Both stocks have the same standard deviation, but stock B is less variable relative to its price

Locating Extreme Outliers: Z-Score To compute the Z-score of a data value, subtract the mean and divide by the standard deviation. The Z-score is the number of standard deviations of the data value above or below the mean. A data value is considered an outlier if its Z-score is less than -3.0 or greater than +3.0. The larger the absolute value of the Z-score, the farther the data value is from the mean. where X represents the data value X is the sample mean S is the sample standard deviation

Locating Extreme Outliers: Z-Score Z-Score is considered an outlier if it is less than -3.0 or greater than +3.0 Suppose the mean math SAT score is 490, with a standard deviation of 100. Compute the Z-score for a test score of 620. Z score of 620 is 1.3 standard deviations above the mean and would not be considered an outlier.

Shape of a Distribution Describes how data are distributed Measures of shape Compare the value between mean and median. Skewness , if skewness is positive, then data skewed to the right; if skewness is negative, then data skewed to the left. Left-Skewed Symmetric Right-Skewed Mean < Median Mean = Median Median < Mean Skewness > 0 Skewness <0 Skewness = 0

General Descriptive Stats Using Microsoft Excel Select Tools. Select Data Analysis. Select Descriptive Statistics and click OK.

General Descriptive Stats Using Microsoft Excel 4. Enter the cell range. 5. Check the Summary Statistics box. 6. Click OK

Excel output Microsoft Excel descriptive statistics output, using the house price data: House Prices: $2,000,000 500,000 300,000 100,000 100,000

Exercise on class TRUE/FALSE QUESTIONS 1. If after graphing the data for a quantitative variable of interest, you notice that the distribution is highly skewed in the positive direction, the measure of central location that would likely provide the best assessment of the center would be the median. Answer: True

2. A statistic is just another name for a parameter. Answer:   False

3. The owner of a local gasoline station has kept track of the number of gallons of regular unleaded sold at his station every day since he purchased the station. This morning, he computed the mean number of gallons. This value would be considered a statistic. Answer: False

4. The Parks and Recreation manager for the city of Detroit recently submitted a report to the city council in which he indicated that a random sample of 500 park users indicated that the average number of visits per month was 4.56. This value should be viewed as a statistic by the city council. Answer: True

5. One of the most frequently used measures of the spread in a set of data is called the mean. Answer: False

6. You are given the following data: 23 34 11 40 25 47  If these data were considered to be a population and you computed the mean, you would get the same answer as if these data were considered to be a sample from a larger population. Answer: True

7. You are given the following data:   23 34 11 40 25 47   Assuming that the data reflect a sample from a larger population, the sample mean is 30.00. Answer: True

8. Data are considered to be right-skewed when the mean lies to the right of the median. Answer: True

9. The mean for a population will generally be larger than the mean from a random sample from that population. Answer: 10. If the population mean is equal to the mode, you can state that the population is symmetric. False False

11. A distribution is said to be symmetric when the sample mean and the population mean are equal. Answer: False

12. In a recent study of the sales prices of houses in a midwestern city, the mean sales price has been reported to be $167,811 while the median sales price was $155,600. From this information, you can determine that the data involved in the study are left-skewed. Answer: False

13. One of the primary advantages of using the median as a measure of the center for a set of data is that the median is not affected by extreme values in the data. Answer: True

14. Suppose a study of houses that have sold recently in your community showed the following frequency distribution for the number of bedrooms: Bedrooms Frequency 1 1 2 18 3 140 4 57 Based on this information, the mode for the data is 140. Answer: False

15. The range is an ideal measure of variation since it is not sensitive to extreme values in the data. Answer: False

16. The Good-Guys Car Dealership has tracked the number of used cars sold at its downtown dealership. Consider the following data as representing the population of cars sold in each of the 8 weeks that the dealership has been open.   3 5 2 7 7 7 9 0 The population range is 3. Answer: False

17. One of the reasons that the standard deviation is preferred as a measure of variation over the variance is that the standard deviation is measured in the original units. Answer: True

18. The standard deviation is a measure of variation of the data around the median. Answer: False

19. For a given set of data, if the data are treated as a population, the calculated standard deviation will be less than it would be had the data been treated as a sample. Answer: True

20. Acme Taxi has two taxi cabs 20. Acme Taxi has two taxi cabs. The manager tracks the daily revenue for each cab. Over the past 20 days, Cab A has averaged $76.00 per night with a standard deviation equal to $11.00. Cab B has averaged $200.00 per night with a standard deviation of $18.00. Based on this information, Cab B has the greatest relative variation. Answer: False

MULTIPLE CHOICE QUESTIONS 1. The most frequently used measure of central tendency is:   a. median. b. mean. c. mode. d. middle value. Answer: b

2. A sample of people who have attended a college football game at your university has a mean = 3.2 members in their family. The mode number of family members is 2 and the median number is 2.0. Based on this information:   a. the population mean exceeds 3.2. b. the distribution is bell-shaped. c. the distribution is right-skewed. d. the distribution is left-skewed. Answer: C

3. A major retail store has studied customer behavior and found that the distribution of time customers spend in a store per visit is symmetric with a mean equal to 17.3 minutes. Based on this information, which of the following is true?   a. The distribution is bell-shaped. b. The median is to the right of the mean. c. The median is approximately 17.3 minutes. d. None of the above. Answer: C

4. Which of the following is the most frequently used measure of variation? a. The range b. The standard deviation c. The variance d. The mode Answer:  b

5. Which of the following measures is not affected by extreme values in the data?   a. The mean b. The median c. The range d. The standard deviation Answer: b

6. The following data reflect the number of customers who test drove new cars each day for a sample of 20 days at the Redfield Ford Dealership.   Given these data, what is the range? a. 14 b. 1 c. Approximately 3.08 d. 5.95 Answer: a

7. Under what circumstances is it necessary to use the coefficient of variation to compare relative variability between two or more distributions? a. When the means of the distributions are equal b. When the means of the distributions are not equal c. When the standard deviations of the distributions are not equal d. When the standard deviations of the distributions are equal Answer: b

Exercise on class SHORT ANSWER QUESTIONS 1. The following is a set of data from a sample of n=6: 7 4 9 7 3 12 a. Compute the mean, median and mode. b. Compute the range, variance, standard deviation and coefficient of variation. c. Compute the Z scores. Are there any outliers? d. Describe the shape of the data set.

Exercise on class 2. The data set contain the starting admission price (in $) for one-day tickets to 10 theme parks in the United States: 58 63 41 42 29 50 62 43 40 40 a. Compute the mean, median and mode. b. Compute the range, variance, and standard deviation. C. Based on the results of (a) and (b), what conclusions can you reach concerning the starting admission price for one-day tickets.

Exercise on class 3. The following is a set of data for population with N=10： 7 5 6 6 6 4 8 6 9 3 a. Compute the population mean. b. Compute the population standard deviation.

Homework Homework next week