Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-1 Business Statistics: A Decision-Making Approach 7 th Edition Chapter 3 Describing Data Using Numerical Measures

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 2-2 Review A frequency histogram is an effective way of converting quantitative data into useful and meaningful information. Visual indication of where data are centered How much spread there is in the data around the center Relative Frequency: when number of total observations of more than two surveys are differ. Simply change frequency to relative frequency (%)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 2-3 Review Grouped Data Frequency Distributions Continuous or discrete data: large number of possible values are spread over a large rage 1. Determine the number of classes or groups 2. Establish class width 3. Determine boundaries

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-4 After completing this chapter, you should be able to: Compute and interpret the mean, median, and mode for a set of data Compute the range, variance, and standard deviation and know what these values mean Construct and interpret a box and whisker graph Compute and explain the coefficient of variation and z scores Use numerical measures along with graphs, charts, and tables to describe data Chapter Goals

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-5 Chapter Topics Measures of Center and Location Mean, median, mode Other measures of Location Weighted mean, percentiles, quartiles Measures of Variation Range, interquartile range, variance and standard deviation, coefficient of variation Using the mean and standard deviation together Coefficient of variation, z-scores

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-6 Summary Measures Center and Location Mean Median Mode Other Measures of Location Weighted Mean Describing Data Numerically Variation Variance Standard Deviation Coefficient of Variation Range Percentiles Interquartile Range Quartiles

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-7 Measures of Center and Location Center and Location MeanMedian ModeWeighted Mean Overview

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-8 Mean (Arithmetic Average) The Mean is the arithmetic average of data values Population mean Sample mean n = Sample Size N = Population Size

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-9 Mean (Arithmetic Average) The most common measure of central tendency Mean = sum of values divided by the number of values Affected by extreme values (outliers) (continued) Mean = Mean = 4

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-10 Median In an ordered array, the median is the “middle” number, i.e., the number that splits the distribution in half The median is not affected by extreme values Useful when data are highly skewed Median = Median = 3

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-11 Median Example Sort the data (array data), note that n = 13 Find the i = (1/2)n position: i = (1/2)(13) = 6.5 Since 6.5 is not an integer, round up to 7 The median is the value in the 7th position: M d = 12 (continued) Data array: 4, 4, 5, 5, 9, 11, 12, 14, 16, 19, 22, 23, 24

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-12 Shape of a Distribution Describes how data is distributed Symmetric or skewed Mean = Median Mean < Median Median < Mean Right-Skewed Left-Skewed Symmetric (Longer tail extends to left)(Longer tail extends to right)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-13 Mode The value that is repeated most often Another measure of central location Not affected by extreme values Used for either numerical or categorical data There may be no mode There may be several modes Mode = No Mode

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-14 Weighted Mean Used when values are grouped by frequency or relative importance (GPA) Days to Complete Frequency Example: Sample of 26 Repair Projects Weighted Mean Days to Complete:

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-15 Five houses on a hill by the beach Review Example House Prices: $2,000, , , , ,000

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-16 Summary Statistics Mean: ($3,000,000/5) = $600,000 Median: middle value of ranked data = $300,000 Mode: most frequent value = $100,000 House Prices: $2,000, , , , ,000 Sum 3,000,000

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-17 Mean is generally used, unless extreme values (outliers) exist Then Median is often used, since the median is not sensitive to extreme values. Example: Realtors say “Median Home Prices”….not average – less sensitive to outliers Which measure of location is the “best”?

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-18 Other Location Measures Other Measures of Location PercentilesQuartiles 1 st quartile = 25 th percentile 2 nd quartile = 50 th percentile = median 3 rd quartile = 75 th percentile The p th percentile in a data array: p% are less than or equal to this value (100 – p)% are greater than or equal to this value (where 0 ≤ p ≤ 100)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-19 Percentiles The p th percentile in an ordered array of n values is the value in i th position, where Example: Find the 60 th percentile in an ordered array of 19 values. If i is not an integer, round up to the next higher integer value So use value in the i = 12 th position

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-20 Quartiles Quartiles split the ranked data into 4 equal groups: Note that the second quartile (the 50 th percentile) is the median 25% Q1Q1 Q2Q2 Q3Q3

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-21 Quartiles Sample Data in Ordered Array: Example: Find the first quartile (n = 9) Q 1 = 25 th percentile, so find i : i = (9) = 2.25 so round up and use the value in the 3 rd position: Q 1 =

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-22 Box and Whisker Plot A box-and-whisker plot (often simply called a box plot) is a graphical way of showing data. It is useful for quickly finding outliers - data points out of line with the rest of the data set. Example: 25% 25% Outliers Lower 1st Median 3rd Upper Limit Quartile Quartile Limit * *

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-23 Shape of Box and Whisker Plots The Box and central line are centered between the endpoints if data is symmetric around the median (A Box and Whisker plot can be shown in either vertical or horizontal format)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-24 Distribution Shape and Box and Whisker Plot Right-SkewedLeft-SkewedSymmetric Q1Q2Q3Q1Q2Q3 Q1Q2Q3

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-25 Constructing the Box and Whisker Plot See the additional note for quartiles on the class website Outliers Lower 1st Median 3rd Upper Limit Quartile Quartile Limit * * The lower limit is Q 1 – 1.5 (Q 3 – Q 1 ) The upper limit is Q (Q 3 – Q 1 ) The center box extends from Q 1 to Q 3 The line within the box is the median The whiskers extend to the smallest and largest values within the calculated limits Outliers are plotted outside the calculated limits

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-26 Box-and-Whisker Plot Example Try this using the online tool on the class website Below is a Box-and-Whisker plot for the following data: This data is right skewed, as the plot depicts Min Q 1 Q 2 Q 3 Max * Upper limit = Q (Q 3 – Q 1 ) = (6 – 2) = is above the upper limit so is shown as an outlier

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-27 Measures of Variation Variation Variance Standard DeviationCoefficient of Variation Population Variance Sample Variance Population Standard Deviation Sample Standard Deviation Range Interquartile Range

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-28 Measures of variation give information on the spread or variability of the data values. Variation Same center, different variation

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-29 Range Simplest measure of variation Difference between the largest and the smallest observations: Range = x maximum – x minimum Range = = 13 Example:

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-30 Ignores the way in which data are distributed Sensitive to outliers Range = = Range = = 5 Disadvantages of the Range 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 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 = = 4 Range = = 119

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-31 Interquartile Range Can eliminate some outlier problems by using the interquartile range Eliminate some high-and low-valued observations and calculate the range from the remaining values. Interquartile range = 3 rd quartile – 1 st quartile

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-32 Interquartile Range Example Median (Q2) X maximum X minimum Q1Q3 Example: 25% 25% Interquartile range = 57 – 30 = 27

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-33 Average of squared deviations of values from the mean Population variance: Sample variance: Variance

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-34 Standard Deviation Most commonly used measure of variation Shows variation about the mean Has the same units as the original data Population standard deviation: Sample standard deviation:

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-35 Calculation Example: Sample Standard Deviation Sample Data (X i ) : n = 8 Mean = x = 16

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-36 Comparing Standard Deviations Mean = 15.5 s = Data B Data A Mean = 15.5 s = Mean = 15.5 s = 4.57 Data C Same mean, but different standard deviations:

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-37 Coefficient of Variation Measures relative variation Always in percentage (%) Shows variation relative to mean Is used to compare two or more sets of data measured in different units Population Sample

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-38 Comparing Coefficients of Variation Stock A: Average price last year = $50 Standard deviation = $5 Stock B: Average price last year = $100 Standard deviation = $5 Both stocks have the same standard deviation, but stock B is less variable relative to its price

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-39 If the data distribution is bell-shaped, then the interval: contains about 68% of the values in the population or the sample The Empirical Rule 68%

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-40 contains about 95% of the values in the population or the sample contains about 99.7% of the values in the population or the sample The Empirical Rule 99.7%95%

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-41 Regardless of how the data are distributed, at least (1 - 1/k 2 ) of the values will fall within k standard deviations of the mean Examples: (1 - 1/1 2 ) = 0% ……..... k=1 (μ ± 1σ) (1 - 1/2 2 ) = 75% … k=2 (μ ± 2σ) (1 - 1/3 2 ) = 89% ………. k=3 (μ ± 3σ) Tchebysheff’s Theorem withinAt least

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-42 A standardized data value refers to the number of standard deviations a value is from the mean Standardized data values are sometimes referred to as z-scores Standardized Data Values

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-43 where: x = original data value μ = population mean σ = population standard deviation z = standard score (number of standard deviations x is from μ) Standardized Population Values

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-44 where: x = original data value x = sample mean s = sample standard deviation z = standard score (number of standard deviations x is from μ) Standardized Sample Values

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-45 IQ scores in a large population have a bell- shaped distribution with mean μ = 100 and standard deviation σ = 15 Find the standardized score (z-score) for a person with an IQ of 121. Someone with an IQ of 121 is 1.4 standard deviations above the mean Standardized Value Example Answer:

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-46 Using Microsoft Excel Descriptive Statistics are easy to obtain from Microsoft Excel Use menu choice: Data / data analysis / descriptive statistics Enter details in dialog box

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-47 Using Excel Select: Data / data analysis / descriptive statistics

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-48 Enter dialog box details Check box for summary statistics Click OK Using Excel (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-49 Excel output Microsoft Excel descriptive statistics output, using the house price data: House Prices: $2,000, , , , ,000

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-50 Chapter Summary Described measures of center and location Mean, median, mode, weighted mean Discussed percentiles and quartiles Created Box and Whisker Plots Illustrated distribution shapes Symmetric, skewed

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-51 Chapter Summary Described measure of variation Range, interquartile range, variance, standard deviation, coefficient of variation Discussed Tchebysheff’s Theorem Calculated standardized data values (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 3-52 Measures of Variation Variation Variance Standard DeviationCoefficient of Variation Population Variance Sample Variance Population Standard Deviation Sample Standard Deviation Range Interquartile Range