1. Edpsy 511
Homework 1: Due 2/6

2. Descriptive Statistics

3. Statistics vs. Parameters
A parameter is a characteristic of a population.
It is a numerical or graphic way to summarize data obtained from the population
A statistic is a characteristic of a sample.
It is a numerical or graphic way to summarize data obtained from a sample

4. Types of Numerical Data
There are two fundamental types of numerical data:
Categorical data: obtained by determining the frequency of occurrences in each of several categories
Quantitative data: obtained by determining placement on a scale that indicates amount or degree

5. Techniques for Summarizing Quantitative Data
Frequency Distributions
Histograms
Stem and Leaf Plots
Distribution curves
Averages
Variability

6. Summary Measures Summary Measures Central Tendency Quartile Variation
Arithmetic Mean
Median
Mode
Range
Variance
Standard Deviation

7. Measures of Central Tendency
Average (Mean)
Median
Mode

8. Mean (Arithmetic Mean)
Mean (arithmetic mean) of data values
Sample mean
Population mean
Sample Size
Population Size

9. Mean The most common measure of central tendency
Affected by extreme values (outliers)
Mean = 5
Mean = 6

10. Mean of Grouped FrequencyX f fX 10 1 9 3 8 2 7 4 6 5
Total N 21

11. Weighted Mean
A form of mean obtained from groups of data in which the different sizes of the groups are accounted for or weighted.

12. Group
xbar N
f(xbar) 1 30 10 2 25 15 3 40

13. Median Robust measure of central tendency
Not affected by extreme values
In an Ordered array, median is the “middle” number
If n or N is odd, median is the middle number
If n or N is even, median is the average of the two middle numbers
Median = 5
Median = 5

14. Mode A measure of central tendency Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data
There may may be no mode
There may be several modes
No Mode
Mode = 9

15. Variability
Refers to the extent to which the scores on a quantitative variable in a distribution are spread out.
The range represents the difference between the highest and lowest scores in a distribution.
A five number summary reports the lowest, the first quartile, the median, the third quartile, and highest score.
Five number summaries are often portrayed graphically by the use of box plots.

16. Variance
The Variance, s2, represents the amount of variability of the data relative to their mean
As shown below, the variance is the “average” of the squared deviations of the observations about their mean
The Variance, s2, is the sample variance, and is used to estimate the actual population variance, s 2

17. Standard Deviation Considered the most useful index of variability.
It is a single number that represents the spread of a distribution.
If a distribution is normal, then the mean plus or minus 3 SD will encompass about 99% of all scores in the distribution.

18. Calculation of the Variance and Standard Deviation of a Distribution (Definitional formula)
Raw
Score Mean X – X (X – X)2
Variance (SD2) =
Σ(X – X)2
N-1
3640 9 =
=404.44 √
Standard deviation (SD) =
Σ(X – X)2
N-1

19. Definitional vs. Computational
An equation that defines a measure
Computational
An equation that simplifies the calculation of the measure

20. Calculate the variance using the computational and definitional formulas.
10, 12, 12, 12, 13, 13, 14

21. In small groups: calculate the variance using the computational and definitional formulas.
2, 8, 9, 11, 15, 17, 20

22. Comparing Standard Deviations
Data A
Mean = 15.5
S = 3.338
Data B
Mean = 15.5
S = .9258
Data C
Mean = 15.5
S = 4.57

23. Facts about the Normal Distribution
50% of all the observations fall on each side of the mean.
68% of scores fall within 1 SD of the mean in a normal distribution.
27% of the observations fall between 1 and 2 SD from the mean.
99.7% of all scores fall within 3 SD of the mean.
This is often referred to as the rule

24. The Normal Curve

25. Different Distributions Compared

26. Fifty Percent of All Scores in a Normal Curve Fall on Each Side of the Mean

27. Probabilities Under the Normal Curve

28. Standard Scores
Standard scores use a common scale to indicate how an individual compares to other individuals in a group.
The simplest form of a standard score is a Z score.
A Z score expresses how far a raw score is from the mean in standard deviation units.
Standard scores provide a better basis for comparing performance on different measures than do raw scores.
A Probability is a percent stated in decimal form and refers to the likelihood of an event occurring.
T scores are z scores expressed in a different form (z score x ).

29. Probability Areas Between the Mean and Different Z Scores

30. Examples of Standard Scores