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