Descriptive Statistics Measures of Central Tendency Variability Standard Scores.
-
Upload
rodrigo-barnett -
Category
Documents
-
view
238 -
download
2
Transcript of Descriptive Statistics Measures of Central Tendency Variability Standard Scores.
![Page 1: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/1.jpg)
Descriptive Statistics
Measures of Central TendencyVariability
Standard Scores
![Page 2: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/2.jpg)
What is TYPICAL???
Average ability conventional circumstances typical appearance most representative ordinary events
![Page 3: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/3.jpg)
Measure of Central Tendency
What SINGLE summary value best describes the central
location of an entire distribution?
![Page 4: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/4.jpg)
Three measures of central tendency (average)
Mode: which value occurs most (what is fashionable)
Median: the value above and below which 50% of the cases fall (the middle; 50th percentile)
Mean: mathematical balance point; arithmetic mean; mathematical mean
![Page 5: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/5.jpg)
Mode For exam data, mode = 37 (pretty
straightforward) (Table 4.1) What if data were
• 17, 19, 20, 20, 22, 23, 23, 28 Problem: can be bimodal, or
trimodal, depending on the scores Not a stable measure
![Page 6: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/6.jpg)
Median For exam scores, Md = 34 What if data were
• 17, 19, 20, 23, 23, 28 Solution:
Best measure in asymmetrical distribution (ie skewed), not sensitive to extreme scores
![Page 7: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/7.jpg)
Nomenclature
X is a single raw score Xi is to the i th score in a set
X n is the last score in a set
Set consists of X 1 , X 2 ,….Xn
X = X 1 + X 2 + …. + X n
![Page 8: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/8.jpg)
Mean
For Exam scores, X = 33.94• Note: X = a single score
Mathematically: X = X / N• the sum of scores divided by the
number of cases• Add up the numbers and divide by
the sample size Try this one: 5,3,2,6,9
![Page 9: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/9.jpg)
Characteristics of the Mean
Balance point•point around which deviation
scores sum to zero
![Page 10: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/10.jpg)
Characteristics of the Mean
Balance point•point around which deviation
scores sum to zero
•Deviation score: Xi - X
•ie Scores 7, 11, 11, 14, 17•X = 12 (X - X) = 0
![Page 11: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/11.jpg)
Balance point Affected by extreme scores
•Scores 7, 11, 11, 14, 17•X = 12, Mode and Median = 11•Scores 7, 11, 11, 14, 170•X = 42.6, Mode & Median = 11
Characteristics of the Mean
Considers value of each individual score
![Page 12: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/12.jpg)
Characteristics of the Mean
Balance point Affected by extreme scores Appropriate for use with
interval or ratio scales of measurement•Likert scale??????????????????
![Page 13: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/13.jpg)
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
![Page 14: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/14.jpg)
Three statisticians out deer hunting
First shoots arrow, sticks in tree to right of the buck
Second shoots arrow, sticks in tree to left of the buck
Third statistician….
![Page 15: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/15.jpg)
More Humour
![Page 16: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/16.jpg)
In Class Assignment
Using the 33 scores that make up exam scores (table 4.1)
students randomly choose 3 scores and calculate mean
WHAT GIVES??
![Page 17: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/17.jpg)
Guidelines to choose Measure of Central Tendency
Mean is preferred because it is the basis of inferential stats•Considers value of each score
![Page 18: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/18.jpg)
Guidelines to choose Measure of Central Tendency
Mean is preferred because it is the basis of inferential stats
Median more appropriate for skewed data??? • Doctor’s salaries• George Will Baseball(1994)• Hygienist’s salaries
![Page 19: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/19.jpg)
To use mean, data distribution must be symmetrical
![Page 20: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/20.jpg)
Normal Distribution
MedianMode
Mean
Scores
![Page 21: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/21.jpg)
Positively skewed distribution
Median
Mode
Mean
Scores
![Page 22: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/22.jpg)
Negatively skewed distribution
![Page 23: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/23.jpg)
Guidelines to choose Measure of Central Tendency
Mean is preferred because it is the basis of inferential statistics
Median more appropriate for skewed data???
Mode to describe average of nominal data (Percentage)
![Page 24: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/24.jpg)
Did you know that the great majorityof people have more than the averagenumber of legs? It's obvious really; amongst the 57 million people in Britainthere are probably 5,000 people who have got only one leg. Therefore the average number of legs is:
![Page 25: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/25.jpg)
Mean = ((5000 * 1) + (56,995,000 * 2)) / 57,000,000 = 1.9999123
Since most people have two legs...
![Page 26: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/26.jpg)
Final (for now) points regarding MCT
Look at frequency distribution•normal? skewed?
Which is most appropiate??
f
Time to fatigue
![Page 27: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/27.jpg)
Alaska’s average elevation of1900 feet is less than that of Kansas. Nothing in that average suggeststhe 16 highest mountains inthe United States are in Alaska. Averages mislead, don’t they?
Grab Bag, Pantagraph, 08/03/2000
![Page 28: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/28.jpg)
Mean may not represent any actual case in the set
Kids Sit up Performance•36, 15, 18, 41, 25
What is the mean? Did any kid perform that many
sit-ups????
![Page 29: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/29.jpg)
Describe the distribution of Japanese
salaries.
![Page 30: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/30.jpg)
Variability defined Measures of Central Tendency provide
a summary level of group performance Recognize that performance (scores)
vary across individual cases (scores are distributed)
Variability quantifies the spread of performance (how scores vary)
parameter or statistic
![Page 31: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/31.jpg)
To describe a distribution
N (n) Measure of Central Tendency
• Mean, Mode, Median Variability
• how scores cluster• multiple measures
• Range, Interquartile range• Standard Deviation
![Page 32: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/32.jpg)
The Range Weekly allowances of son & friends
• 2, 5, 7, 7, 8, 8, 10, 12, 12, 15, 17, 20
Everybody gets $12; Mean = 10.25
![Page 33: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/33.jpg)
The Range Weekly allowances of son & friends
• 2, 5, 7, 7, 8, 8, 10, 12, 12, 15, 17, 20 Range = (Max - Min) Score
• 20 - 2 = 18 Problem: based on 2 cases
![Page 34: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/34.jpg)
The Range Allowances
• 2, 5, 7, 7, 8, 8, 10, 12, 12, 15, 17, 20
Susceptible to outliers Allowances
• 2, 2, 2, 3, 4, 4, 5, 5, 5, 6, 7, 20 Range = 18 Mean = 5.42
Mean = 10.25
Outlier
![Page 35: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/35.jpg)
Semi-Interquartile range
What is a quartile??
![Page 36: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/36.jpg)
What is a quartile??•Divide sample into 4 parts
•Q1 , Q2 , Q3 => Quartile Points
Interquartile Range = Q 3 - Q 1
SIQR = IQR / 2 Related to the Median
Calculate with atable12.sav data, output on next overhead
Semi-Interquartile range
![Page 37: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/37.jpg)
Case Summariesa
Ted 2.00 2.00
Mary 5.00 2.00
Bob 7.00 2.00
Lou 7.00 3.00
Marge 8.00 4.00
Sue 8.00 4.00
Leo 10.00 5.00
Kate 12.00 5.00
Moe 12.00 5.00
Phil 15.00 6.00
Zeke 17.00 7.00
Zach 20.00 20.00
12 12 12
1
2
3
4
5
6
7
8
9
10
11
12
NTotal
NAME TEST1 TEST2
Limited to first 100 cases.a.
Ata
ble
12.s
av
![Page 38: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/38.jpg)
Quartiles of Test 1 & Test 2(Procedure Frequencies on SPSS)
Statistics
12 12
0 0
7.0000 2.2500
9.0000 4.5000
14.2500 5.7500
Valid
Missing
N
25
50
75
Percentiles
TEST1 TEST2
Calculate inter-quartile range for Test 1 and Test 2
![Page 39: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/39.jpg)
BMD and walkingQuartiles based on miles walked/week
Krall et al, 1994, Walking is related to bone density and rates of bone loss. AJSM, 96:20-26
![Page 40: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/40.jpg)
Standard Deviation
Statistic describing variation of scores around the mean
Recall concept of deviation score
![Page 41: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/41.jpg)
Standard Deviation
Statistic describing variation of scores around the mean
Recall concept of deviation score•DS = Score - criterion score•x = Raw Score - Mean
What is the sum of the x’s?
![Page 42: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/42.jpg)
Standard Deviation
Statistic describing variation of scores around the mean
Recall concept of deviation score•DS = Score - criterion score•x = Raw Score - Mean
What is the mean of the x’s?
![Page 43: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/43.jpg)
Standard Deviation
Statistic describing variation of scores around the mean
Recall concept of deviation score•x = Raw Score - Mean x2
Variance = N Average squared deviation score
![Page 44: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/44.jpg)
Problem
Variance is in units squared, so inappropriate for description
Remedy???
![Page 45: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/45.jpg)
Standard Deviation
Take the square root of the variance
square root of the average squared deviation from the mean x2
SD = N
![Page 46: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/46.jpg)
TOP TEN REASONS TO BECOME A STATISTICIAN
Deviation is considered normal.We feel complete and sufficient.We are "mean" lovers.Statisticians do it discretely and continuously.We are right 95% of the time.We can legally comment on someone's posterior distribution.We may not be normal but we are transformable.We never have to say we are certain.We are honestly significantly different.No one wants our jobs.
![Page 47: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/47.jpg)
Calculate Standard Deviation
Use as scores1, 5, 7, 3
Mean = 4 Sum of deviation scores = 0
(X - X)2 = 20• read “sum of squared deviation scores”
Variance = 5 SD = 2.24
![Page 48: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/48.jpg)
Key points about deviation scores
If a deviation score is relatively small, case is close to mean
If a deviation score is relatively large, case is far from the mean
![Page 49: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/49.jpg)
Key points about SD SD small data clustered round mean SD large data scattered from the mean Affected by extreme scores (as per mean) 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)
![Page 50: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/50.jpg)
Reporting descriptive statistics in a paper
Descriptive statistics for vertical ground reaction force (VGRF) are presented in Table 3, and graphically in Figure 4. The mean (± SD) VGRF for the experimental group was 13.8 (±1.4) N/kg, while that of the control group was 11.4 (± 1.2) N/kg.
![Page 51: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/51.jpg)
Figure 4. Descriptive statistics of VGRF.
0
5
10
15
20
Exp Con
![Page 52: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/52.jpg)
SD and the normal curve
60 70 80
X = 70SD = 10 34% 34%
About 68% ofscores fallwithin 1 SDof mean
![Page 53: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/53.jpg)
The standard deviation and the normal curve
About 68% ofscores fallbetween 60 and 70
60 70 80
X = 70SD = 10
34% 34%
![Page 54: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/54.jpg)
The standard deviation and the normal curve
70
About 95% ofscores fallwithin 2 SDof mean
60 8050 90
X = 70SD = 10
![Page 55: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/55.jpg)
70
About 95% ofscores fallbetween 50 and 90
60 8050 90
X = 70SD = 10
The standard deviation and the normal curve
![Page 56: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/56.jpg)
The standard deviation and the normal curve
70
About 99.7% of scores fall within 3 S.D. of the mean
60 8050 90
X = 70SD = 10
40 100
![Page 57: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/57.jpg)
The standard deviation and the normal curve
70
About 99.7% of scores fall between 40 and 100
60 8050 90
X = 70SD = 10
40 100
![Page 58: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/58.jpg)
What about X = 70, SD = 5?
What approximate percentage of scores fall between 65 & 75?
What range includes about 99.7% of all scores?
![Page 59: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/59.jpg)
Descriptive statistics for a normal population
n Mean SDAllows you to formulate the limits (range) includinga certain percentage (Y%) of all scores.Allows rough comparison of different sets of scores.
More on the SD and the Normal Curve
![Page 60: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/60.jpg)
Comparing Means Relevance of
Variability
![Page 61: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/61.jpg)
Effect SizeMean Difference as % of SD
Small: 0.2 SDMedium: 0.5 SDLarge: 0.8 SD
Cohen (1988)
![Page 62: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/62.jpg)
Male &
Female Strength
![Page 63: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/63.jpg)
Pooled Standard Deviation
If two samples have similar, but not identical standard deviations
SS1 + SS2
Sdpooled= n1 + n2
or Sd1 + Sd2
Sdpooled~ 2
![Page 64: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/64.jpg)
Male &
Female Strength
Sdpooled = 198+340 2 = 269
Mean Difference = 416-942 = -526
Effect Size = -526/269 = -1.96
![Page 65: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/65.jpg)
ABOUT
Area under Normal Curve• Specific SD values (z) including
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%)
http://psych.colorado.edu/~mcclella/java/normal/tableNormal.html
Quebec Hydro article
![Page 66: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/66.jpg)
Descriptive Statistics
51 32.665 18.116
51
(cents/pack)
Valid N (listwise)
N Mean Std. Deviation
What upper and lower limitsinclude 95% of scores?
![Page 67: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/67.jpg)
Standard Scores
Comparing scores across (normal) distributions • “z-scores”
![Page 68: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/68.jpg)
Assessing the relative position of a single score
Move from describing a distribution to looking at how a single score fits into the group•Raw Score: a single individual value
•ie 36 in exam scores
How to interpret this value??
![Page 69: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/69.jpg)
Descriptive Statistics
Mean SD n
Describe the “typical” and the “spread”, and the number of cases
![Page 70: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/70.jpg)
Descriptive Statistics
Mean SD n
Describe the “typical” and the “spread”, and the number of cases
z-score•identifies a score as above or below the mean AND expresses a score in units of SD
• z-score = 1.00 (1 SD above mean)• z-score = -2.00 (2 SD below mean)
![Page 71: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/71.jpg)
Z-score = 1.0GRAPHICALLY
Z = 1
84% of scores smaller than this
![Page 72: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/72.jpg)
Calculating z-scores
Z = X - XSD
Calculate Z for each of the following situations: 32,3,20 XSDX
6,2,9 XSDX
DeviationScore
![Page 73: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/73.jpg)
Other features of z-scores
Mean of distribution of z-scores is equal to 0 (ie 0 = 0 SD)
Standard deviation of distribution of z-scores = 1•since SD is unit of measurement
z-score distribution is same shape as raw score distribution
data from atable41.sav
![Page 74: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/74.jpg)
Z-scores: allow comparison of scores from different distributions
Mary’s score• SAT Exam 450 (mean 500 SD 100)
Gerald’s score• ACT Exam 24 (mean 18 SD 6)
Who scored higher?
Mary: (450 – 500)/100 = - .5Gerald: (24 – 18)/6 = 1
![Page 75: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/75.jpg)
Interesting use of z-scores: Compare performance on
different measures
ie Salary vs Homeruns•MLB (n = 22, June 1994)
•Mean salary = $2,048,678• SD = $1,376,876
•Mean HRs = 11.55• SD = 9.03
•Frank Thomas•$2,500,000, 38 HRs
![Page 76: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/76.jpg)
More z-score & bell-curve
For any z-score, we can calculate the percentage of scores between it and the mean of the normal curve; between it and all scores below; between it and all scores above• Applet demos:
• http://psych.colorado.edu/~mcclella/java/normal/normz.html• http://psych.colorado.edu/~mcclella/java/normal/handleNormal.html• http://psych.colorado.edu/~mcclella/java/normal/tableNormal.html
![Page 77: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/77.jpg)
Recall, when z-score = 1.0 ...
50%
34.13%
![Page 78: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/78.jpg)
% scores above z = 1.0
50%
34.13%
15.87%
![Page 79: Descriptive Statistics Measures of Central Tendency Variability Standard Scores.](https://reader036.fdocuments.net/reader036/viewer/2022081506/56649c915503460f9494b981/html5/thumbnails/79.jpg)
If z-score = 1.2
X 1.2 SD
50%
What % in here?