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Measures of Central Measures of Central Tendency and VariabilityTendency and Variability
Chapter 5:113-123
Using Normal Curves For Evaluation
Types of Curves...
The Normal Curve:
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”
When “Average” isn’t When “Average” isn’t Good EnoughGood EnoughRepresentative“Normal”“Typical”Not
Outstanding or Extreme
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.
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
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…
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
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
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)
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
Did you hear the one about the two
statisticians who went pheasant hunting
together?
The Point PleaseThe “Cluster”
of a set of scores is one thing
Spread may actually be more important for interpretation
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
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
Curve “Skewness”
MeanMedian
More than ½ the scoresAre above the mean:Skewed Left
More than ½ the scores Are below the mean:Skewed Right
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
Example: Population: PE MajorsSample: PE Majors at WSC,
graduating in 2002Measurement: Mean Starting
SalaryResults: $78,000
–Believe it?
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!!
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!
Evaluating Individual Scores
Normal Curves
Z-Scores
Comparing Apples to Oranges…
Use of “Group” StatisticsCompare
different groups
Evaluate individuals within the group
QUESTION: “What if your roommate came home and said, “I got a 95 on my test!” ?
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
Individuals want to know what their scores mean. They want some kind of a judgment so they can make decisions.
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
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
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
What the heck does that mean?Z-Scores
reflect a score’s relationship to the rest of the scores....
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....
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!
Conclusions:
His score was only better than ~2.5% of all students (that’s bad)How did I get there?
Graphing the Data:
9896.5 99.595 101
68%
95%
2.5%
SummarySummaryMeans and Standard Deviations
describe groups of scoresNormal curves have predictable
dimensionsZ-Scores convert raw scores into
multiples of the standard deviation
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
Assignment: ProblemsCalculating Z Scores:
Determine the MeanDetermine the SDThe Z score for ANY
INDIVIDUAL in that group is calculated:
Zi = (Xi – X)/ SD