1. Descriptive Statistics

2. the everyday notions of central tendency
Usual
Customary
Most
Standard
Expected
normal
Ordinary
Medium
commonplace
NY Times, 10/24/ 2010
Stories vs. Statistics
By JOHN ALLEN PAULOS

3. Overview What are descriptive statistics?
A bit of terminology/notation
Measures of Central Tendency
Mean, Mode, Median
Measures of Variability
Ranges, Standard Deviations
The Normal Curve

4. Terminology/Notation
A data distribution = A set of data/scores (the whole thing)
1, 2, 4, 7
X = A raw, single score (i.e., 2 from above)
∑ = Summation (added up)
∑X = 14 (each individual score added up)
n = sample size (distribution size, or number of scores)
n = 4 (from above)

5. Descriptive Statistics
Descriptive statistics are the side of statistics we most often use in our everyday lives
Realize that most observations/data are too “large” for a human to take in and comprehend – we must “reduce” them
How can we summarize what we see?
Example – Grades/Registrar

6. Descriptive Statistics
Descriptive statistics = describing the data
n = 50, a test score of 83%
Where does it fit in the class??
Making sense out of chaos

7. Descriptive Statistics
Transform a set of numbers or observations into indices that describe or characterize the data
“Summary statistics”
A large group of statistics that are used in all research manuscripts
Even the most complex statistical tests and studies start with descriptive statistics

8. Descriptive Statistics
Measurement
Scales
Relationship
Scatterplot
Correlation
Regression
Nominal
Ordinal
Interval
Ratio
Descriptive Statistics
Graphic
Portrayals
Variability
Central
Tendency
Range
Standard deviation
Standardized scores
Frequencies
Histograms
Bar graphs
Normal distribution
Mean
Median
Mode

9. Descriptive Statistics
Descriptive statistics usually accomplish two major goals:
1) Describe the central location of the data
2) Describe how the data are dispersed about that point
In other words, they provide:
1) Measures of Central Tendency
2) Measures of Variability

10. Measure of Central Tendency
What SINGLE summary value best describes the CENTRAL location of an entire distribution?
Mode: which value occurs most often
Median: the value above and below which 50% of the cases fall (the middle; 50th percentile)
Mean: mathematical balance point; arithmetic/mathematical average

11. Mode Most frequent occurrence What if data were?
17, 19, 20, 20, 22, 23, 25, 28
17, 19, 20, 20, 22, 23, 23, 28
Problem: set of numbers can be bimodal, or trimodal, depending on the scores
Not a stable measure
Ex. 17, 19, 20, 22, 23, 28, 28

12. Median Rank numbers, pick middle one What if data were…?
17, 19, 20, 23, 23, 28
Solution: add up two middle scores, divide by 2 (=21.5)
Best measure in asymmetrical distribution (i.e. skewed), not sensitive to extreme scores
Ex. 17, 19, 20, 23, 23, 428

13. Mean = X
Add up the numbers and divide by the sample size (the number of numbers!)
Try this one…
2,3,5,6,9
= 25 / 5 = 5
(Usually) best measure of the three –uses the most information (all values from distribution contribute)

14. Characteristics of the Mean
Balance point
Point around which deviations sum to zero
Deviation = X – X
For instance, if scores are 2,3,5,6,9
Mean is 5
Sum of deviations: (-3)+(-2)+0+1+4=0
∑ (X – X) = 0

15. Characteristics of the Mean
Affected by extreme scores
Example 1
Scores 7, 11, 11, 14, 17
Mean = 12, Mode and Median = 11
Example 2
Scores 7, 11, 11, 14, 170
Mean = 42.6, Mode & Median = 11

16. Characteristics of the Mean
Balance point
Affected by extreme scores
Appropriate for use with interval or ratio scales of measurement
More stable than Median or Mode when multiple samples drawn from the same population
Basis for inferential stats

17. Guidelines to Choose Measure of Central Tendency
Mean is preferred because it is the basis of inferential statistics
Median may be better for skewed data
Distribution of wealth in the US – ex. annual household income in Washington state for 2000: mean=$76,818; median=$42,024
Mode to describe average of nominal data (eye color, hair color, etc…)

18. Normal Distribution
Frequency,
How often a score occurs
Scores

19. MLB batting averages over 3-year span (min. 100 AB)
Mean = 0.267
n = 1291

20. Normal Distribution Scores Mode
“Normal” distribution indicates the data are perfectly symmetrical
Median
Mean
Scores

21. Positively skewed distribution
Mode
Median
Mean
Scores

22. NFL Salaries 2011

23. Negatively skewed distribution
Mode
Median
Mean
Scores

24. Relationship among the MCT & shape of distribution

25. Alaska’s average elevation of 1900 feet is less than that of Kansas.
Nothing in that average suggests
the 16 highest mountains in
the United States are in Alaska.
Averages mislead, don’t they?
Grab Bag, Pantagraph, 08/03/2000

26. Measures of dispersion or spread
Variability
Measures of dispersion or spread
The only thing constant is variation.

27. the notions of variability
Unusual
Peculiar
Strange
Original
Extreme
Special
Unlike
Deviant
Dissimilar
different
NY Times, 10/24/ 2010
Stories vs. Statistics
By JOHN ALLEN PAULOS

28. Variability defined
Measures of Central Tendency provide a summary level of the data
Recognizes that scores vary across individual cases
ie, the mean or median may not be an actual score in your distribution
Variability quantifies the spread of performance
How scores vary around mean/mode/median

29. To describe a distribution
1) Measure of Central Tendency
Mean, Mode, Median
2) Measure of Variability
Multiple measures
Range, Interquartile range, Semi-Interquartile Range
Standard Deviation

30. Range Range = Difference between low/high score
# of hours spent watching TV/week
2, 5, 7, 7, 8, 8, 10, 12, 12, 15, 17, 20
Range = (Max - Min) Score
= 18
Very susceptible to outliers
Doesn’t indicate anything about variability around the mean/central point

31. Semi-Interquartile range
What is a quartile??
Divide sample into 4 parts of equal size
Q1 , Q2 , Q3 = Quartile Points
Interquartile Range = Q3 - Q1
Difference between highest and lowest quartile
SIQR = IQR / 2
Related to the Median…prevents outliers from overly skewing measure
For ordinal data or skewed interval/ratio

32. Quartiles based on miles walked/week
BMD and walking
Quartiles based on miles walked/week
Krall et al, 1994, Walking is related to bone density and rates of bone loss. AJSM, 96:20-26

33. Notes:
Skewed Distribution?
95th Percentile?
50th Percentile vs Median?

34. Standard Deviation Most commonly accepted measure of spread
Variation itself is nature's only irreducible essence.
Stephen Jay Gould
Standard Deviation
Most commonly accepted measure of spread
Compute the deviations of all numbers from the mean
Square and THEN sum each of the deviations
Divide by the number of deviations
Finally, take the square root

35. Standard Deviation Distribution = 1, 3, 5, 7 X = 16 /4 = 4
1) Compute Deviations = -3, -1, 1, 3
2) Square Deviations = 9, 1, 1, 9
3) Sum Deviations = 20
4) Divide by n= 20/4 = 5
5) Take square root = √5 = 2.2

36. Key points about SD SD small  data clustered round mean
SD largedata scattered from the mean
Affected by extreme scores (just like mean)…oftentimes called “outliers”
Consistent (more stable) across samples from the same population
Just like the mean - so it works well with inferential stats (where repeated samples are taken)

37. SD Example Three NFL quarterbacks with similar QB ratings in 2006:
Matt Hasselbeck (SEA) = 76.0
Rex Grossman (CHI) = 73.9
Brett Favre (GB) = 72.7
Note: QB rating involves a complex formula accounting for passing attempts, completions, yards, touchdowns, and interceptions…100+ is considered outstanding & is average
All appear to have had very similar, somewhat mediocre seasons as QB’s

38. SD Example Let’s look at the SD of their game-by-game QB ratings:
Matt Hasselbeck (SEA) = 29.97
Rex Grossman (CHI) = 47.60
Brett Favre (GB) = 27.81
Grossman had, by far, the most variability (i.e. inconsistency) in his game-by-game performances…is this good or bad?

39. Clinical Use of SD

40. SD and the normal curve
The following concepts are critical to your understanding of how descriptive statistics works
Remember – a “normal” curve is perfectly symmetrical. This is not typical, but usually data are almost normal…

41. SD and the normal curve X = 70 SD = 10 34.1% 34.1% About 68% of
scores fall
within 1 SD
of mean
X = 70
SD = 10
34.1%
34.1% 60 70 80

42. The standard deviation and the normal curve
About 68% of
scores fall
between 60 and 70
X = 70
SD = 10
34%
34% 60 70 80

43. The standard deviation and the normal curve
About 95% of
scores fall
within 2 SD
of mean
X = 70
SD = 10
34.1%
34.1%
13.6%
13.6% 50 60 70 80 90

44. The standard deviation and the normal curve
About 95% of
scores fall
between 50 and 90
X = 70
SD = 10
34.1%
34.1%
13.6%
13.6% 50 60 70 80 90

45. The standard deviation and the normal curve
About 99.7% of scores fall within 3 S.D. of the mean
X = 70
SD = 10
34.1%
34.1%
13.6%
13.6%
2.3%
2.3% 40 50 60 70 80 90
100

46. The standard deviation and the normal curve
About 99.7% of scores fall between 40 and 100
X = 70
SD = 10
34.1%
34.1%
13.6%
13.6%
2.3%
2.3% 40 50 60 70 80 90
100

47. What about = 70, SD = 5?
What approximate percentage of scores fall between 65 & 75?
…1SD below + 1SD above = 68%
What range includes about 99.7% of all scores?
…3SD below to 3SD above = 55 to 85

48. Interpreting The Normal Table
Area under Normal Curve
Specific SD values (z) include certain percentages of the scores
Values of Special Interest
1.96 SD = 47.5% of scores ( = 95%)
2.58 SD = 49.5% of scores ( = 99%)
ie, 95% of scores fall within 1.96 standard deviations of the mean (1.96 above and 1.96 below)

49. IQ X = 100 SD = 15 68% have an IQ between 85-115 55 70 85 100 115 130
34.1%
34.1%
13.6%
13.6%
2.3%
2.3% 55 70 85
100
115
130
145

50. MLB players’ batting averages over a 3-year span (min. 100 at bats)
~95% of players have an average between and 0.337

51. Next Week…
We will utilize our understanding of descriptive statistics concepts, including central tendency, variability, and the normal curve, to examine standardized scores
Homework = Cronk 3.1 – 3.4
Bring calculator to class
In-class activity 2…