1. Descriptive Statistics
Notes from
McMillan & Schumacher

2. Statistics
• Methods of organizing and analyzing
quantitative data
• Tools designed to help researcher organize
and interpret numbers derived from
measuring a trait or variable
• International language that only manipulates
numbers
• Numbers do not interpret themselves
• Meaning derived from the research design

3. Categories of Statistics
Descriptive and Inferential
Descriptive Statistics
• Summarize, organize, and reduce large
numbers of observations to a few numbers
• Describe or characterize the data
• Assigning numbers to things in order to
differentiate one thing from another

4. Inferential Statistics
• Use to make inferences or predictions from the
sample to the population
• Depend on descriptive statistics
• Estimation of population characteristics from
sample
• Experimental Designs
– True Experimental – random assignment
– Quasi Experimental – Pre-test/post-test, time series
– Single subject

5. Descriptive to Inferential
Descriptive Statistics
Of Sample
Population
Sample
Estimate
Population
Based on
Descriptive
Statistics

6. Scales of Measurement
Nominal – Categories (sex, ethnicity, party)
Ordinal – Rank order (percentile norms)
Interval – Equal difference between #s
Ratio – Equal amounts from zero

7. Nominal Scale of Measurement
• Nominal - Name
• Categories and Classifications
• Naming of mutually exclusive categories
• People, events, phenomena
• Eye color, gender, political party, group
• No order or value implied
• Assign number for coding - arbitrary
• Numbers do not represent quantity or
degree

8. Ordinal Scale of Measurement
• Ordinal – by ranked order
• Categories rank-ordered from highest to
lowest
– Equal =
– Greater than >
– Less than <
• Ranking by grade point average, percentile,
achievement test score, social class

9. Interval Scale of Measurement
• Shares characteristics with ordinal
• Equal intervals between each category
• Equal difference between variables or
attributes being measures
• Constant unit of measurement
• Difference between 5&6 = 18&19
• Year, Centigrade, Fahrenheit

10. Ratio Scale of Measurement
• Most refined type of measurement
• Ordinal and Interval
• Numbers can be compared by ratios
• Numbers represent equal amounts from
absolute zero
• Age, dollars, time, speed, class size
• Most educational measurement – not ratio

11. Graphic Portrayals of Data
• Frequency Distribution – f
• Number of times each score was attained
• Rank order and then tally
• Show most/least occurring scores
• General Shape of Distribution
• Outliers

12. Histograms and Bar Graphs
• Graphic way of
representing
frequency
distribution
• Histogram –
frequencies rank-
ordered
• Bar Graph –
order arbitrary

13. Frequency Polygon
• Illustrates
frequency
distribution
• Single points
rather than bars
are graphed
• Normal curve –
curves the
straight lines

14. Measures of Central Tendency
• Mean –
– arithmetic average of all scores
• Median –
– point which divides a rank-ordered distribution
into halves that have an equal number of scores
– 50% above and 50% below
• Mode – score that occurs most frequently

15. Relationships among Measures
of Central Tendency
• Normal Distribution
– Mean, median, and mode about the same
– Bell shaped symmetrical curve
– Large numbers – normal distribution
• Skewed Distribution
– Positively skewed – Most scores at low end
– Negatively skewed – Most scores at high end
– Lower numbers distributed unevenly

16. Normal Distribution

17. Bell Curve – Normal Distribution
Mean = Median = Mode0 100
Normal curve – theoretical distribution used to
transform data and calculate many statistics

18. Positively Skewed
Mode Mean Median0 100
Most of the
scores are at the
low end of the
distribution

19. Negatively Skewed
Mean Median Mode
0 100
Most of the
scores are at the
high end of the
distribution

20. Measures of Variability
• Shows how spread out the distribution of the
scores is from the mean of the distribution –
dispersion of scores
• How much, on average, does each score
differ from the mean?
• Variability measures
– Range – highest and lowest (no mean)
– Standard Deviation – numerical index indicating
average variability of scores

21. Standard Deviation
• Indicates the amount on average that the set
of scores deviates from the mean

22. SD in Normal Distribution
-1 SD0 100
34% 34%
+1 SD
68%
+1 SD = 84th
percentile
-1 SD = 16th
percentile
50%
below the
mean
50% above
the mean

23. Box and Whisker Plot
• Use to give
picture of
variability
• Size of
rectangular box
– 25th
to 75th
percentiles
• Whiskers draw
from ends of
box to 10th
and
90th
percentiles

24. Standard Scores
• Makes it easier to analyze several
distributions if means and standard
deviations are different for each distribution
• Raw scores converted to standard scores
• Provide constant normative or relative
meaning
• Obtained from the mean and standard
deviation of the raw score distribution

25. The Z-Score
• Most basic standard score
• Mean of 0
• Standard deviation of 1
• Z-score of +1 = 84th
percentile
• Z-score of –1 = 16th
percentile
• Example – IQ tests
100 = mean
15/16 = standard deviation

26. Scatterplot
• Graphic
representation of
relationship of
variables
• Relationships
– Positive
– Negative
– None
– Curvilinear

27. Correlation Coefficient
• Calculated number representing the
relationships between variables
• Range from –1.00 to +1.00
• High Positive Relationship (.85 .90. 96)
• Low Positive Relationship (.15 .20 . 08)
- 1 +10
High negative High positive

28. Types of Correlation Coefficients
• See Table 7.5 – page 172
• Most common
– Pearson product-moment
• r
• Both continuous
– Spearman
• rs
• Both rank-ordered

29. Example of Correlation - SPSS