Measures of Central Tendency and Variability Chapter 5:113-123 Using Normal Curves For Evaluation.
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Transcript of Measures of Central Tendency and Variability Chapter 5:113-123 Using Normal Curves For Evaluation.
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Measures of Central Measures of Central Tendency and VariabilityTendency and Variability
Chapter 5:113-123
Using Normal Curves For Evaluation
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Types of Curves...
The Normal Curve:
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Normal Means “Average” …Normal Means “Average” …Sort ofSort of In a Normal Distribution, most of
the scores are found closest to the middleThey’re “average”
Either “tail” represents rare scores They’re “special”
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When “Average” isn’t When “Average” isn’t Good EnoughGood EnoughRepresentative“Normal”“Typical”Not
Outstanding or Extreme
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Statistical Measures of Statistical Measures of Central TendencyCentral TendencyMean: The calculated “average”Median: The middle of the
ordered scoresMode: The most frequently
occurring score(s)
The mean is the measure of choice ifYou want to do further statistical analysis.
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The MeanThe Mean
X = Σxi / N
Considered more precise and stable than the median or mode
Can be used in additional statistical analysis
Don’t use with nominal or ordinal data
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The MedianThe Median In an ORDERED set of scoresThe Median score is exactly in
the middleMedian = MdnMdn = (Number of scores +1)/ 2That tells us where the Mdn
score is found…
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Like so:Like so:Set of scores: 5, 6, 3, 7, 4, 9, 2
Order the scores: 2, 3, 4, 5, 6, 7, 9
Find the position of the median Score: Mdn = (N+1) / 2 Mdn = (7+1) / 2 = 4
The median score is the 4th score: 2, 3, 4, 5, 6, 7, 9
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Comparing the Median Comparing the Median and Mean Scores:and Mean Scores: Mdn = 5X = 36/7 =
5.14Make a
conclusion about this set of scores
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The Mode:The Mode:The most frequently occurring
score(s)Gives a quick BUT ROUGH sense of
the typical score…Can you think of a situation when the
MODE is not the mean or median, but is a better description of what the typical student in your group is like? (HINT: Lab 1)
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Pull-Up ScoresPull-Up ScoresPullups
0
5
10
15
0-2 3-5 6-8 9-11 12-14 15-17 18-21 21-23 24-25
Number of pullups
Num
ber
of
Stu
dent
s
X = 4.8 pull-ups
The mode is usually used to describe the most typical score in NOMINAL data: Eg. Nebraska is the most commonbirth-state of WSC students
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Did you hear the one about the two
statisticians who went pheasant hunting
together?
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The Point PleaseThe “Cluster”
of a set of scores is one thing
Spread may actually be more important for interpretation
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What is the Standard What is the Standard Deviation?Deviation?The appropriate measure of the
variability of a set of scores, when the mean is used as the measure of central tendency.
The average deviation of any randomly chosen score from the mean
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Using the Mean and Median Using the Mean and Median to determine “Normalcy”to determine “Normalcy” 50% of the scores fall above and below the
Median score It will be exactly in the middle of the range
of scores When the Mean = Median, the curve is
NORMAL When Mean > Median it is skewed Right When Mean < Median it is skewed left…like
so
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Curve “Skewness”
MeanMedian
More than ½ the scoresAre above the mean:Skewed Left
More than ½ the scores Are below the mean:Skewed Right
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Why the Fuss About Normal Why the Fuss About Normal Curves?Curves?Whole populations will always be
distributed in a “Normal” arrangement
For a SAMPLE of that population to accurately reflect the population, the sample MUST BE NORMAL – or conclusions won’t be valid
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Example: Population: PE MajorsSample: PE Majors at WSC,
graduating in 2002Measurement: Mean Starting
SalaryResults: $78,000
–Believe it?
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WSC PE Graduates: Salary
<$20k $25-29K >$40K
2
6
8
N = 20Range: $12,500 - $350,000Mean: $78,000SD: +/- $52,000
This guy plays For the NBA andMakes $350K!!
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The Truth:The Truth:If we through out the NBA
player, the mean is then $29,050
With the NBA player in there…the mean is “skewed to the right” of the true average of the “typical” graduate…
BUYER BEWARE!
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Evaluating Individual Scores
Normal Curves
Z-Scores
Comparing Apples to Oranges…
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Use of “Group” StatisticsCompare
different groups
Evaluate individuals within the group
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QUESTION: “What if your roommate came home and said, “I got a 95 on my test!” ?
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What does his score What does his score mean?mean?
There were 200 possibleThe highest score was only 101The mean was 98The range was 95-101
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Individuals want to know what their scores mean. They want some kind of a judgment so they can make decisions.
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Types of Norm Referenced EvaluationsPercentile Rank:
mathematically tedious, defined as the percent of the scores below an individuals score
Z-Scores: Calculating how many standard deviations a score is from the mean
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A Word About Percentile A Word About Percentile Ranks: Ranks: Compares your score to the rest of
the “group”Norm-Referenced EvaluationBUT WHAT GROUP?
National Norms: ACT scores, President’s Fitness Test
Local Norms: Developed from at least 100 local scores
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Calculating Z-ScoresCalculating Z-ScoresFind the mean and standard
deviation of a set of scoresZi
= (Xi - X)/ s
The value of Z is a multiple +/- of the standard deviation
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What the heck does that mean?Z-Scores
reflect a score’s relationship to the rest of the scores....
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Let’s Jump to Let’s Jump to ConclusionsConclusions
-Z = below average+Z = above averageValue of Z = how many standard
deviations (How far below)68% of the scores will be within 1
standard deviation....
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Let’s Evaluate yourLet’s Evaluate your Roommate’s Score by Z-ScoreRoommate’s Score by Z-Score:
Mean = 98SD = 1.5XR = 95ZR = (XR – X)/ SD
Z = (95-98)/1.5 = -2 Your roommate’s score is 2 standard
deviations below average!
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Conclusions:
His score was only better than ~2.5% of all students (that’s bad)How did I get there?
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Graphing the Data:
9896.5 99.595 101
68%
95%
2.5%
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SummarySummaryMeans and Standard Deviations
describe groups of scoresNormal curves have predictable
dimensionsZ-Scores convert raw scores into
multiples of the standard deviation
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Summary cont.
Finally: Using Z-scores to evaluate (give meaning to) an individual’s score is a type of Norm Referenced Evaluation
Z-Scores can only be used in “Normal” groups
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Assignment: ProblemsCalculating Z Scores:
Determine the MeanDetermine the SDThe Z score for ANY
INDIVIDUAL in that group is calculated:
Zi = (Xi – X)/ SD