1. Introduction to Descriptive Statistics Objectives: 1.Explain the general role of statistics in assessment & evaluation 2.Explain three methods for describing a data set: shape, center, and spread 3.Explain the relationship between the standard deviation and the normal curve “Data have a story to tell. Statistical analysis is detective work in which we apply our intelligence and our tools to discover parts of that story.”-Hamilton (1990)

2. Levels of Measurement Nominal Nominal Ordinal Ordinal Interval Interval Ratio Ratio Determining what statistics are appropriate Determining what statistics are appropriate

3. Nominal Naming things. Naming things. Creating groups that are qualitatively different or unique… Creating groups that are qualitatively different or unique… But not necessarily quantitatively different. But not necessarily quantitatively different.

4. Nominal Placing individuals or objects into categories. Placing individuals or objects into categories. Making mutually excusive categories. Making mutually excusive categories. Numbers assigned to categories are arbitrary. Numbers assigned to categories are arbitrary.

5. Nominal Sample variables: Sample variables: – Gender – Race – Ethnicity – Geographic location – Hair or eye color

6. Ordinal Rank ordering things. Rank ordering things. Creating groups or categories when only rank order is known. Creating groups or categories when only rank order is known. Numbers imply order but not exact quantity of anything. Numbers imply order but not exact quantity of anything.

7. Ordinal The difference between individuals with adjacent ranks, on relevant quantitative variables, is not necessarily the same across the distribution. The difference between individuals with adjacent ranks, on relevant quantitative variables, is not necessarily the same across the distribution.

8. Ordinal Sample variables: Sample variables: – Class Rank – Place of finish in a race (1 st, 2 nd, etc.) – Judges ratings – Responses to Likert scale items (for example – SD, D, N, A, SA)

9. Interval Orders observations according to the quantity of some attribute. Orders observations according to the quantity of some attribute. Arbitrary origin. Arbitrary origin. Equal intervals. Equal intervals. Equal differences expressed as equal distances. Equal differences expressed as equal distances.

10. Interval Sample variables: Sample variables: – Test Scores SAT SAT GRE GRE IQ tests IQ tests – Temperature Celsius Celsius Fahrenheit Fahrenheit

11. Ratio Quantitative measurement. Quantitative measurement. Equal intervals. Equal intervals. True zero point. True zero point. Ratios between values are useful. Ratios between values are useful.

12. Ratio Sample variables: Sample variables: – Financial variables – Finish times in a race – Number of units sold – Test scores scaled as percent correct or number correct

13. Levels of Measurement Review What level of measurement? What level of measurement? – Today is a fall day. – Today is the third hottest day of the month. – The high today was 70 o Fahrenheit. – The high today was 20 o Celsius. – The high today was 294 o Kelvin.

14. Levels of Measurement Review What level of measurement? What level of measurement? – Student #1256 is: – a male – from Lawrenceville, GA. – He came in third place in the race today. – He scored 550 on the SAT verbal section. – He has turned in 8 out of the 10 homework assignments.

15. Levels of Measurement Review What level of measurement? What level of measurement? – Student #3654 is: – in the third reading group. – Nominal? – Ordinal? – Interval? – Ratio?

16. Descriptive Statistics Used to describe the basic features of a batch of data. Uses graphical displays and descriptive quantitative indicators. The purpose of descriptive statistics is to organize and summarize data so that the data is more readily comprehended. That is, descriptive statistics describes distributions with numbers.

17. Five Descriptive Questions What is the middle of the set of scores? What is the middle of the set of scores? How spread out are the scores? How spread out are the scores? Where do specific scores fall in the distribution of scores? Where do specific scores fall in the distribution of scores? What is the shape of the distribution? What is the shape of the distribution? How do different variables relate to each other? How do different variables relate to each other?

18. Five Descriptive Questions Middle Middle Spread Spread Rank or Relative Position Rank or Relative Position Shape Shape Correlation Correlation

19. Middle Mean Mean Median Median Mode Mode

20. Examples of these measures Mean of: 2, 3, 6, 7, 3, 5, 10 Mean of: 2, 3, 6, 7, 3, 5, 10 (2 + 3 + 6 + 7 + 3 + 5 + 10)/ 7 = 36/ 7 = 5.14 Mode of: 2, 3, 6, 7, 3, 5, 10 is 3 Mode of: 2, 3, 6, 7, 3, 5, 10 is 3 Median of: 2, 3, 6, 7, 3, 5, 10 Median of: 2, 3, 6, 7, 3, 5, 10 First data is ordered: 2, 3, 3, 5, 6, 7, 10. Middle value is 5 therefore that is the median.

21. Some Important Points Mode is the only descriptive measure used for nominal data Mode is the only descriptive measure used for nominal data Median is unaffected by extreme values, it is resistant to extreme observations. Median is unaffected by extreme values, it is resistant to extreme observations. Mean or Average is affected by extremely small or large values. We say that it is sensitive or nonresistant to the influence of extreme observations. The mean is the balance point of the distribution. Mean or Average is affected by extremely small or large values. We say that it is sensitive or nonresistant to the influence of extreme observations. The mean is the balance point of the distribution. In symmetric distributions the mean and median are close together. In symmetric distributions the mean and median are close together.

22. More important points In skewed data the mean is pulled to the tail of the distribution. In skewed data the mean is pulled to the tail of the distribution. Median is not necessarily preferred over the mean even if it is resistant. However if data is known to be strongly skewed then the median is preferable. Median is not necessarily preferred over the mean even if it is resistant. However if data is known to be strongly skewed then the median is preferable. Finally, the average is usually the measurement of central tendency of choice because it is stable during sampling. Finally, the average is usually the measurement of central tendency of choice because it is stable during sampling.

23. Spread Standard Deviation Standard Deviation Variance Variance Range Range IQR IQR

24. Large S Average or Normal S Small S X = Mean S = Standard Deviation X = 50 (S = 20) X = 50 (S = 10) X = 50 (S = 5) How do measures of variability differ when distributions are spread out? Describing Data: Center & Spread

25. Rank or Relative Position Five number summary Five number summary Min, 25 th, 50 th, 75 th, Max Min, 25 th, 50 th, 75 th, Max Identifying specific values that have interpretive meaning Identifying specific values that have interpretive meaning Identifying where they fall in the set of scores Identifying where they fall in the set of scores Box plots Box plots Outliers Outliers

26. Shape Positive Skewness Positive Skewness Negative Skewness Negative Skewness Normality Normality Histograms Histograms

27. Shape - Normality

28. Shape- Positive Skewness

29. Shape – Negative Skewness

30. When a distribution of data resembles a normal distribution (or normal curve):  68% of the data lies within + or – 1 standard deviation  95% of the data lie within + or – 2 standard deviations  99.7% of the data lie within + or – 3 standard deviations from the mean 99.7% 95% 68% Relating the Standard Deviation (S) to the normal distribution. “68-95-99.7% Rule” Describing Data: Center & Spread

31. Outliers

32. Outliers

33. Outliers