Describing Data Using Statistics Statistics and Risk Management Copyright © Texas Education Agency, All rights reserved.1

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Tools of Knowledge Statistics : The study of collecting, organizing, and analyzing data. For Research For Research For Reporting For Reporting Copyright © Texas Education Agency, All rights reserved.3

When Did This Come About? We find Statistical Information (State of Things) in We find Statistical Information (State of Things) in By the 18th century, the term "statistics" designated the systematic collection of demographic and economic data by states. By the 18th century, the term "statistics" designated the systematic collection of demographic and economic data by states. In the early 19th century, the meaning of "statistics" broadened, then including the discipline concerned with the collection, summary, and analysis of data. In the early 19th century, the meaning of "statistics" broadened, then including the discipline concerned with the collection, summary, and analysis of data. Copyright © Texas Education Agency, All rights reserved.4

Vocabulary to Know Population: is the over all group of subjects to which the research will apply. Population: is the over all group of subjects to which the research will apply. Sample: is a smaller, more practical, group size that represents the larger population. Sample: is a smaller, more practical, group size that represents the larger population. Copyright © Texas Education Agency, All rights reserved.5 Subjects:

Research Statistics? Descriptive Statistics: Procedures used to organize and present data in a convenient and communicable form. Descriptive Statistics: Procedures used to organize and present data in a convenient and communicable form. Inferential Statistics: Procedures employed arrive at broader conclusions or inferences about populations on the basis of samples. Inferential Statistics: Procedures employed arrive at broader conclusions or inferences about populations on the basis of samples. Copyright © Texas Education Agency, All rights reserved.6

Data Types Qualitative Data Qualitative Data Quantitative Data Quantitative Data Discrete data Discrete data Continuous data Continuous data Copyright © Texas Education Agency, All rights reserved.7

Sample Measurements Count: Total Number of Scores in a set Count: Total Number of Scores in a set Low/High: The Extreme Values-lowest and highest in a set Low/High: The Extreme Values-lowest and highest in a set Range: From the Low to the High-the distance between the lowest and the highest. For example if my lowest number in the set was 2 and my highest number in the set was 20, my range would be 18. Range: From the Low to the High-the distance between the lowest and the highest. For example if my lowest number in the set was 2 and my highest number in the set was 20, my range would be = = 18 Copyright © Texas Education Agency, All rights reserved.8

The Count = 24 9:00 pm210 9:00 pm210 9:00 am205 9:00 am205 1:00 pm200 1:00 pm200 6:00 pm180 6:00 pm180 10:00 am170 10:00 am170 12:00 am170 12:00 am170 2:00 pm170 2:00 pm170 10:00 pm161 10:00 pm161 8:00 am 160 8:00 am 160 3:00 pm152 3:00 pm152 11:00 am150 11:00 am150 8:00 pm 143 8:00 pm 143 4:00 pm140 4:00 pm140 5:00 pm130 5:00 pm130 6:00 am120 6:00 am120 7:00 am110 7:00 am110 7:00 pm100 7:00 pm100 11:00 pm80 11:00 pm80 12:00 am40 12:00 am40 2:00 am40 2:00 am40 1:00 am30 1:00 am30 3:00 am20 3:00 am20 4:00 am10 4:00 am10 5:00 am31 5:00 am31 Copyright © Texas Education Agency, All rights reserved.9

The High = 210 9:00 pm210 9:00 pm210 9:00 am205 9:00 am205 1:00 pm200 1:00 pm200 6:00 pm180 6:00 pm180 10:00 am170 10:00 am170 12:00 am170 12:00 am170 2:00 pm170 2:00 pm170 10:00 pm161 10:00 pm161 8:00 am 160 8:00 am 160 3:00 pm152 3:00 pm152 11:00 am150 11:00 am150 8:00 pm 143 8:00 pm 143 4:00 pm140 4:00 pm140 5:00 pm130 5:00 pm130 6:00 am120 6:00 am120 7:00 am110 7:00 am110 7:00 pm100 7:00 pm100 11:00 pm80 11:00 pm80 12:00 am40 12:00 am40 2:00 am40 2:00 am40 1:00 am30 1:00 am30 3:00 am20 3:00 am20 4:00 am10 4:00 am10 5:00 am31 5:00 am31 Copyright © Texas Education Agency, All rights reserved.10

The Low = 31 9:00 pm210 9:00 pm210 9:00 am205 9:00 am205 1:00 pm200 1:00 pm200 6:00 pm180 6:00 pm180 10:00 am170 10:00 am170 12:00 am170 12:00 am170 2:00 pm170 2:00 pm170 10:00 pm161 10:00 pm161 8:00 am 160 8:00 am 160 3:00 pm152 3:00 pm152 11:00 am150 11:00 am150 8:00 pm 143 8:00 pm 143 4:00 pm140 4:00 pm140 5:00 pm130 5:00 pm130 6:00 am120 6:00 am120 7:00 am110 7:00 am110 7:00 pm100 7:00 pm100 11:00 pm80 11:00 pm80 12:00 am40 12:00 am40 2:00 am40 2:00 am40 1:00 am30 1:00 am30 3:00 am20 3:00 am20 4:00 am10 4:00 am10 5:00 am31 5:00 am31 Copyright © Texas Education Agency, All rights reserved.11

The Range = 179 9:00 pm210 9:00 pm210 9:00 am205 9:00 am205 1:00 pm200 1:00 pm200 6:00 pm180 6:00 pm180 10:00 am170 10:00 am170 12:00 am170 12:00 am170 2:00 pm170 2:00 pm170 10:00 pm161 10:00 pm161 8:00 am 160 8:00 am 160 3:00 pm152 3:00 pm152 11:00 am150 11:00 am150 8:00 pm 143 8:00 pm 143 4:00 pm140 4:00 pm140 5:00 pm130 5:00 pm130 6:00 am120 6:00 am120 7:00 am110 7:00 am110 7:00 pm100 7:00 pm100 11:00 pm80 11:00 pm80 12:00 am40 12:00 am40 2:00 am40 2:00 am40 1:00 am30 1:00 am30 3:00 am20 3:00 am20 4:00 am10 4:00 am10 5:00 am31 5:00 am31 What about a sample with an odd number of scores? Copyright © Texas Education Agency, All rights reserved. 12

Measures of Central Tendencies Mode: The most repeated score Mode: The most repeated score Median: The score(s) in the middle Median: The score(s) in the middle Mean: The average of all scores Mean: The average of all scores Copyright © Texas Education Agency, All rights reserved.13 Let’s take a look at a more descriptive reference of these three measures of central tendencies…

The Median: ( )/2= :00 pm210 9:00 pm210 9:00 am205 9:00 am205 1:00 pm200 1:00 pm200 6:00 pm180 6:00 pm180 10:00 am170 10:00 am170 12:00 am170 12:00 am170 2:00 pm170 2:00 pm170 10:00 pm161 10:00 pm161 8:00 am 160 8:00 am 160 3:00 pm152 3:00 pm152 11:00 am150 11:00 am150 8:00 pm 143 8:00 pm 143 4:00 pm140 4:00 pm140 5:00 pm130 5:00 pm130 6:00 am120 6:00 am120 7:00 am110 7:00 am110 7:00 pm100 7:00 pm100 11:00 pm80 11:00 pm80 12:00 am40 12:00 am40 2:00 am40 2:00 am40 1:00 am30 1:00 am30 3:00 am20 3:00 am20 4:00 am10 4:00 am10 5:00 am31 5:00 am31 What about a sample with an odd number of scores? Copyright © Texas Education Agency, All rights reserved. 14

The Mode 6:00 am120 6:00 am120 7:00 am110 7:00 am110 8:00 am 160 8:00 am 160 9:00 am205 9:00 am205 10:00 am170 10:00 am170 11:00 am150 11:00 am150 12:00 am170 12:00 am170 1:00 pm200 1:00 pm200 2:00 pm170 2:00 pm170 3:00 pm152 3:00 pm152 4:00 pm140 4:00 pm140 5:00 pm130 5:00 pm130 6:00 pm180 6:00 pm180 7:00 pm100 7:00 pm100 8:00 pm 143 8:00 pm 143 9:00 pm210 9:00 pm210 10:00 pm161 10:00 pm161 11:00 pm80 11:00 pm80 12:00 am40 12:00 am40 1:00 am30 1:00 am30 2:00 am40 2:00 am40 3:00 am20 3:00 am20 4:00 am10 4:00 am10 5:00 am31 5:00 am31 Copyright © Texas Education Agency, All rights reserved.15

The Mean (Average) Mean (Average) Number of Scores Each ScoreSum Notation: Sample data uses CAPITAL Roman letters Copyright © Texas Education Agency, All rights reserved.16

Spreadsheet Formulas =Sum(B2:B11)/Count(B2:B11)Or=Average(B2:B11) Copyright © Texas Education Agency, All rights reserved.17

Not Enough!! Why? Sample OneSample Two Total: 26.9 Mean: 5.38 Copyright © Texas Education Agency, All rights reserved.18

Deviation from the Mean Sample OneSample Two = = = = = = = = = = 5.12 Tot. Deviation: 0.0 Mean: 5.38 That does not work! Copyright © Texas Education Agency, All rights reserved. 19

Let’s Square Them Sample OneSample Two ( ) 2 =.14( ) 2 = 6.86 ( ) 2 = 1.04( ) 2 = ( ) 2 =.14( ) 2 =.38 ( ) 2 =.77( ) 2 = ( ) 2 =.38( ) 2 = Tot.Variance: 2.47Tot.Variance: Mean: 5.38 Works, but values are exaggerated! Copyright © Texas Education Agency, All rights reserved. 20

Standard Deviation Take the total variances and divide them by N-1. Then take the square root of that. Sample OneSample Two (2.47/(5-1))^.5 =.79(68.47/(5-1))^.5 = 4.14 Standard Deviation of the Sample Sample MeanScore Number of Scores The X subscript only shows the S is for the sample Do you sum the differences then square them or square the differences then sum them? Copyright © Texas Education Agency, All rights reserved.21

Degrees of Freedom Why do we divide by N-1 instead of just N? N-1 is referred to as Degrees of Freedom. It was decided that df is used because it makes the resulting amount larger by using a consistent methodology….that’s all. You will see it again. Copyright © Texas Education Agency, All rights reserved.22

Review Questions Copyright © Texas Education Agency, All rights reserved.23 On a sheet of notebook paper (or you can type these up on the computer), answer the following questions. You may use the PowerPoint presentations, as well as websites and any notes you have acquired to answer the questions, just make sure you complete these ON YOUR OWN! 1.What are some of the main types of CHARTS? 2.When and how are those charts normally applied? 3.How important are indices? 4.Describe what statistics is. 5.How can statistics be used to describe a list of data? 6.What are the elements of Measuring Central Tendencies of a data list? 7.Explain Variance measurements. 8.What do we mean when we refer to Degrees of Freedom?

Student Assignments Go to my webpage for statistics: – Click on the link “Student Activity 3.2a” and complete the assignment. When you have finished, go on to “Student Activity 3.2b” and complete it as well. Copyright © Texas Education Agency, All rights reserved.24