Basic Stat Notes

Graphs, Charts, and Tables – Describing Your Data Business Statistics Dr.M.Raghunadh Acharya 06/08/09

Contents … ,[object Object],[object Object],[object Object],[object Object],06/08/09

Frequency Distributions ,[object Object],[object Object],[object Object],[object Object],06/08/09

Why Use Frequency Distributions? ,[object Object],[object Object],[object Object],06/08/09

Frequency Distribution: Discrete Data ,[object Object],Example: An advertiser asks 200 customers how many days per week they read the daily newspaper. 06/08/09 Number of days read Frequency 0 44 1 24 2 18 3 16 4 20 5 22 6 26 7 30 Total 200

Relative Frequency ,[object Object],22% of the people in the sample report that they read the newspaper 0 days per week 06/08/09 Number of days read Frequency Relative Frequency 0 44 .22 1 24 .12 2 18 .09 3 16 .08 4 20 .10 5 22 .11 6 26 .13 7 30 .15 Total 200 1.00

Frequency Distribution: Continuous Data ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/08/09

Grouping Data by Classes ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/08/09

Frequency Distribution Example Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Class Frequency 10 but under 20 3 .15 20 but under 30 6 .30 30 but under 40 5 .25 40 but under 50 4 .20 50 but under 60 2 .10 Total 20 1.00 Relative Frequency Frequency Distribution 06/08/09

Histograms ,[object Object],[object Object],[object Object],[object Object],06/08/09

Histogram Example Class Midpoints Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 No gaps between bars, since continuous data 06/08/09

Questions for Grouping Data into Classes ,[object Object],[object Object],[object Object],[object Object],[object Object],06/08/09

How Many Class Intervals? ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],(X axis labels are upper class endpoints) 06/08/09

General Guidelines ,[object Object],[object Object],[object Object],[object Object],06/08/09

Class Width ,[object Object],[object Object],Largest Value  Smallest Value Number of Classes W = 06/08/09

Histograms in Excel ,[object Object],[object Object],1 06/08/09

[object Object],2 3 Input data and bin ranges Select Chart Output Histograms in Excel (continued) 06/08/09

Stem and Leaf Diagram ,[object Object],[object Object],06/08/09

Example: ,[object Object],Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 ,[object Object],[object Object],Stem Leaf 1 2 3 5 06/08/09

Example: ,[object Object],Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 06/08/09 Stem Leaves 1 2 3 7 2 1 4 4 6 7 8 3 0 2 5 7 8 4 1 3 4 6 5 3 8

Using other stem units ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Stem Leaf 06/08/09

Graphing Categorical Data Categorical Data Pie Charts Pareto Diagram Bar Charts 06/08/09

Bar and Pie Charts ,[object Object],[object Object],06/08/09

Pie Chart Example Percentages are rounded to the nearest percent Current Investment Portfolio Savings 15% CD 14% Bonds 29% Stocks 42% Investment Amount Percentage Type (in thousands $) Stocks 46.5 42.27 Bonds 32.0 29.09 CD 15.5 14.09 Savings 16.0 14.55 Total 110 100 (Variables are Qualitative) 06/08/09

Pareto Diagram Example cumulative % invested (line graph) % invested in each category (bar graph) 06/08/09

Bar Chart Example 06/08/09 Number of days read Frequency 0 44 1 24 2 18 3 16 4 20 5 22 6 26 7 30 Total 200

Tabulating and Graphing Multivariate Categorical Data ,[object Object],Investment Investor A Investor B Investor C Total Category Stocks 46.5 55 27.5 129 Bonds 32.0 44 19.0 95 CD 15.5 20 13.5 49 Savings 16.0 28 7.0 51 Total 110.0 147 67.0 324 06/08/09

Tabulating and Graphing Multivariate Categorical Data ,[object Object],(continued) 06/08/09

Side-by-Side Chart Example ,[object Object],06/08/09

[object Object],[object Object],[object Object],[object Object],Line Charts and Scatter Diagrams 06/08/09

Line Chart Example 06/08/09 Year Inflation Rate 1985 3.56 1986 1.86 1987 3.65 1988 4.14 1989 4.82 1990 5.40 1991 4.21 1992 3.01 1993 2.99 1994 2.56 1995 2.83 1996 2.95 1997 2.29 1998 1.56 1999 2.21 2000 3.36 2001 2.85 2002 1.58

Scatter Diagram Example 06/08/09 Volume per day Cost per day 23 125 26 140 29 146 33 160 38 167 42 170 50 188 55 195 60 200

Types of Relationships ,[object Object],06/08/09

[object Object],Types of Relationships (continued) 06/08/09

Chapter Summary ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/08/09

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Summarization measures ….. 06/08/09

Center and Location Mean Median Mode Other Measures of Location Weighted Mean Describing Data Numerically Variation Variance Standard Deviation Coefficient of Variation Range Percentiles Inter quartile Range Quartiles Summary Measures 06/08/09

Center and Location Mean Median Mode Weighted Mean Overview: Measures of Center and Location 06/08/09

[object Object],[object Object],[object Object],n = Sample Size N = Population Size Mean (Arithmetic Average) 06/08/09

[object Object],[object Object],[object Object],0 1 2 3 4 5 6 7 8 9 10 Mean = 3 0 1 2 3 4 5 6 7 8 9 10 Mean = 4 Mean (Arithmetic Average) 06/08/09

[object Object],[object Object],[object Object],[object Object],0 1 2 3 4 5 6 7 8 9 10 Median = 3 0 1 2 3 4 5 6 7 8 9 10 Median = 3 Median 06/08/09

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 5 0 1 2 3 4 5 6 No Mode Mode 06/08/09

[object Object],Example : Sample of 26 Repair Projects Weighted Mean Days to Complete: Weighted Mean 06/08/09 Days to Complete Frequency 5 4 6 12 7 8 8 2

[object Object],House Prices: $2,000,000 500,000 300,000 100,000 100,000 Review Example 06/08/09

[object Object],[object Object],[object Object],[object Object],House Prices: $2,000,000 500,000 300,000 100,000 100,000 Sum 3,000,000 Summary Statistics 06/08/09

[object Object],[object Object],[object Object],Which measure of location is the “best”? 06/08/09

[object Object],[object Object],Mean = Median = Mode Mean < Median < Mode Mode < Median < Mean Right-Skewed Left-Skewed Symmetric (Longer tail extends to left) (Longer tail extends to right) Shape of a Distribution 06/08/09

[object Object],[object Object],[object Object],[object Object],Other Measures of Location Percentiles Quartiles ,[object Object],[object Object],[object Object],[object Object],Other Location Measures 06/08/09

[object Object],[object Object],Percentiles 06/08/09

[object Object],25% 25% 25% 25% Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22 ,[object Object],Q1 Q2 Q3 Quartiles (n = 9) Q1 = 25 th percentile, so find the so use the value half way between the 2 nd and 3 rd values, so 25 100 (9+1) = 2.5 position 25 100 Q1=12.5 06/08/09

[object Object],Minimum -- Q1 -- Median -- Q3 -- Maximum Example : 25% 25% 25% 25% Box and Whisker Plot 06/08/09

[object Object],[object Object],Shape of Box and Whisker Plots 06/08/09

Right-Skewed Left-Skewed Symmetric Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3 Distribution Shape and Box and Whisker Plot 06/08/09

[object Object],[object Object],0 2 3 5 27 Min Q1 Q2 Q3 Max Box-and-Whisker Plot Example 06/08/09

Variation Variance Standard Deviation Coefficient of Variation Population Variance Sample Variance Population Standard Deviation Sample Standard Deviation Range Interquartile Range Measures of Variation 06/08/09

[object Object],Same center, different variation Variation 06/08/09

[object Object],Range = x maximum – x minimum 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Range = 14 - 1 = 13 Example: Range 06/08/09

7 8 9 10 11 12 Range = 12 - 7 = 5 7 8 9 10 11 12 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 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 = 5 - 1 = 4 Range = 120 - 1 = 119 Disadvantages of the Range ,[object Object],[object Object],06/08/09

[object Object],[object Object],[object Object],Interquartile Range 06/08/09

Median (Q2) X maximum X minimum Q1 Q3 Example : 25% 25% 25% 25% 12 30 45 57 70 Interquartile range = 57 – 30 = 27 Interquartile Range 06/08/09

[object Object],[object Object],[object Object],Variance 06/08/09

[object Object],[object Object],[object Object],[object Object],[object Object],Standard Deviation 06/08/09

Sample Data (X i ) : 10 12 14 15 17 18 18 24 n = 8 Mean = x = 16 Calculation Example: Sample Standard Deviation 06/08/09

Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21 11 12 13 14 15 16 17 18 19 20 21 Data B Data A Mean = 15.5 s = .9258 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = 4.57 Data C Comparing Standard Deviations 06/08/09

[object Object],[object Object],[object Object],[object Object],Population Sample Coefficient of Variation 06/08/09

[object Object],[object Object],[object Object],Both stocks have the same standard deviation, but stock B is less variable relative to its price Comparing Coefficient of Variation ,[object Object],[object Object],[object Object],06/08/09

[object Object],[object Object],The Empirical Rule X 68% 06/08/09

[object Object],[object Object],The Empirical Rule 99.7% 95% 06/08/09

[object Object],[object Object],[object Object],[object Object],[object Object],within At least Tchebysheff’s Theorem 06/08/09

[object Object],[object Object],Standardized Data Values 06/08/09

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Standardized Population Values 06/08/09

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Standardized Sample Values 06/08/09

Basic Stat Notes

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