1. Introduction to Educational Statistics
Joseph Stevens, Ph.D., University of Oregon
(541) ,

2. WHAT IS STATISTICS?
Statistics is a group of methods used to collect, analyze, present, and interpret data and to make decisions.

3. POPULATION VERSUS SAMPLE
A population consists of all elements – individuals, items, or objects – whose characteristics are being studied. The population that is being studied is also called the target population.

4. POPULATION VERSUS SAMPLE cont.
The portion of the population selected for study is referred to as a sample.

5. POPULATION VERSUS SAMPLE cont.
A study that includes every member of the population is called a census. The technique of collecting information from a portion of the population is called sampling.

6. POPULATION VERSUS SAMPLE cont.
A sample drawn in such a way that each element of the population has an equal chance of being selected is called a simple random sample.

7. TYPES OF STATISTICS
Descriptive Statistics consists of methods for organizing, displaying, and describing data by using tables, graphs, and summary measures.

8. TYPES OF STATISTICS
Inferential Statistics consists of methods that use information from samples to make predictions, decisions or inferences about a population.

9. Basic Definitions
A variable is a characteristic under study that assumes different values for different elements. A variable on which everyone has the same exact value is a constant.

10. Basic Definitions
The value of a variable for an element is called an observation or measurement.

11. Basic Definitions
A data set is a collection of observations on one or more variables.
A distribution is a collection of observations or measurements on a particular variable.

12. TYPES OF VARIABLES Quantitative Variables
Discrete Variables
Continuous Variables
Qualitative or Categorical Variables

13. Quantitative Variables cont.
A variable whose values are countable is called a discrete variable. In other words, a discrete variable can assume only a limited number of values with no intermediate values.

14. Quantitative Variables cont.
A variable that can assume any numerical value over a certain interval or intervals is called a continuous variable.

15. Categorical Variables
A variable that cannot assume a numerical value but can be classified into two or more categories is called a categorical variable.

16. Scales of Measurement
How much information is contained in the numbers?
Operational Definitions and measurement procedures
Types of Scales
Nominal
Ordinal
Interval
Ratio

17. Descriptive Statistics
Variables can be summarized and displayed using:
Tables
Graphs and figures
Statistical summaries:
Measures of Central Tendency
Measures of Dispersion
Measures of Skew and Kurtosis

18. Measures of Central Tendency
Mode – The most frequent score in a distribution
Median – The score that divides the distribution into two groups of equal size
Mean – The center of gravity or balance point of the distribution

19. Median
The calculation of the median consists of the following two steps:
Rank the data set in increasing order
Find the middle number in the data set such that half of the scores are above and half below. The value of this middle number is the median.

20. Arithmetic Mean
The mean is obtained by dividing the sum of all values by the number of values in the data set.
Mean for sample data:

21. Example: Calculation of the mean
Four scores: 82, 95, 67, 92

22. The Mean is the Center of Gravity95 92 82 67

23. The Mean is the Center of Gravity
X (X – X)
82 82 – 84 = -2
95 95 – 84 = +11
67 67 – 84 = -17
92 92 – 84 = +8
∑(X – X) = 0

24. Comparison of Measures of Central Tendency

25. Measures of Dispersion
Range
Variance
Standard Deviation

26. Range
Highest value in the distribution minus the lowest value in the distribution + 1

27. Variance
Measure of how different scores are on average in squared units:
∑(X – X)2 / N

28. Standard Deviation Returns variance to original scale units
Square root of variance = sd

29. Other Descriptors of Distributions
Skew – how symmetrical is the distribution
Kurtosis – how flat or peaked is the distribution

30. Kinds of Distributions
Uniform
Skewed
Bell-shaped or Normal
Ogive or S-shaped

32. Normal distribution with mean μ and standard deviation σx

33. Total area under a normal curve.
The shaded area is 1.0 or 100% μ x

34. A normal curve is symmetric about the mean
Each of the two shaded areas is .5 or 50% .5 .5 μ x

35. Areas of the normal curve beyond μ ± 3σ.
Each of the two shaded areas is very close to zero
μ – 3σ μ
μ + 3σ x

36. Three normal distribution curves with the same mean but different standard deviations
σ = 5
σ = 10
σ = 16 x
μ = 50

37. Three normal distributions with different means but the same standard deviation
σ = σ = σ = 5
µ = µ = µ = x

38. Areas under a normal curve
For a normal distribution approximately
68% of the observations lie within one standard deviation of the mean
95% of the observations lie within two standard deviations of the mean
99.7% of the observations lie within three standard deviations of the mean

39. μ – 3σ μ – 2σμ – σ μ μ + σ μ + 2σ μ + 3σ
99.7%
95%
68%
μ – 3σ μ – 2σμ – σ μ μ + σ μ + 2σ μ + 3σ

40. Score Scales Raw Scores Percentile Ranks Grade Equivalents (GE)
Standard Scores
Normal Curve Equivalents (NCE)
Z-scores
T-scores
College Board Scores

42. Converting an X Value to a z Value
For a normal random variable X, a particular value of x can be converted to its corresponding z value by using the formula
where μ and σ are the mean and standard deviation of the normal distribution of x, respectively.

43. The Logic of Inferential Statistics
Population: the entire universe of individuals we are interested in studying
Sample: the selected subgroup that is actually observed and measured (with sample size N)
Sampling Distribution of a Statistic: a distribution of samples like ours

44. The Three Distributions Used in Inferential Statistics
I. Population
III. Sampling Distribution of the Statistic
II. Sample