Chapter 2 Means to an End:

Computing and Understanding Averages

Part IISigma Freud & Descriptive

Statistics

What you will learn in Chapter 2

Measures of central tendency

Computing the mean for a set of scores

Computing the mode and median

Selecting a measure of central tendency

Measures of Central Tendency

The AVERAGE is a single score that best represents a set of scores

Averages are also know as “Measures of Central Tendency”

Three different ways to describe the distribution of a set of scores…Mean – typical average scoreMedian – middle scoreMode – most common score

Computing the Mean

Formula for computing the mean

“X bar” is the mean value of the group of scores

“” (sigma) tells you to add together whatever follows it

X is each individual score in the groupThe n is the sample size

XX

n

Things to remember…

N = population n = sampleSample mean is the measure of central

tendency that best represents the population mean

Mean is VERY sensitive to extreme scores that can “skew” or distort findings

Average means the one measure that best represents a set of scoresDifferent types of averagesType of average used depends on the data

and question

Weighted Mean Example

List all values for which the mean is being calculated (list them only once)

List the frequency (number of times) that value appears

Multiply the value by the frequencySum all Value x FrequencyDivide by the total Frequency (total n size)

Computing the Median

Median = point/score at which 50% of remaining scores fall below and 50% fall above.

NO standard formulaRank order scores from highest to lowest or

lowest to highestFind the “middle” score

BUT…What if there are two middle scores?What if the two middle scores are the same?

A little about Percentiles…

Percentile points are used to define the percent of cases equal to and below a certain point on a distribution75th %tile – means that the score received is

at or above 75 % of all other scores in the distribution

“Norm referenced” measureallows you to make comparisons

Computing the Mode

Mode = most frequently occurring scoreNO formula

List all values in the distributionTally the number of times each value occursThe value occurring the most is the mode

Democrats = 90Republicans = 70Independents = 140: the MODE!!

When two values occur the same number of times -- Bimodal distribution

When to Use What…

Use the Mode when the data are categorical

Use the Median when you have extreme scores

Use the Mean when you have data that do not include

extreme scores and are not categorical

Using SPSS

Glossary Terms to Know

AverageMeasures of Central Tendency

MeanWeighted meanArithmetic mean

MedianPercentile pointsOutliers

Mode

Chapter 3 Viva La Difference: Understanding

Variability

Part IISigma Freud & Descriptive

Statistics

What you will learn in Chapter 3

Variability is valuable as a descriptive toolDifference between variance & standard

deviationHow to compute:

RangeStandard DeviationVariance

Why Variability is Important

VariabilityHow different scores are from one particular

scoreSpreadDispersion

What is the “score” of interest here?Ah ha!! It’s the MEAN!!

So…variability is really a measure of how each score in a group of scores differs from the mean of that set of scores.

Measures of Variability

Three types of variability that examine the amount of spread or dispersion in a group of scores…Range Standard DeviationVariance

Typically report the average and the variability together to describe a distribution.

Computing the Range

Range is the most “general” estimate of variability…

Two types…Exclusive Range

R = h - lInclusive Range

R = h – l + 1

(Note: R is the range, h is the highest score, l is the lowest score)

Computing Standard Deviation

Standard Deviation (SD) is the most frequently reported measure of variability

SD = average amount of variability in a set of scores

What do these symbols represent?

Why n – 1?

The standard deviation is intended to be an estimate of the POPULATION standard deviation…We want it to be an “unbiased estimate”Subtracting 1 from n artificially inflates the

SD…making it largerIn other words…we want to be

“conservative” in our estimate of the population

Things to Remember…

Standard deviation is computed as the average distance from the mean

The larger the standard deviation the greater the variability

Like the mean…standard deviation is sensitive to extreme scores

If s = 0, then there is no variability among scores…they must all be the same value.

Computing Variance

Variance = standard deviation squared

So…what do these symbols represent? Does the formula look familiar?

Standard Deviation or Variance

While the formulas are quite similar…the two are also quite different.Standard deviation is stated in original unitsVariance is stated in units that are squared

Which do you think is easier to interpret???

Using the Computer to Compute Measures of Variability

Glossary Terms to Know

VariabilityRangeStandard deviation

Mean deviationUnbiased estimate

Variance

Chapter 4 A Picture is Really Worth a Thousand

Words

Part IISigma Freud & Descriptive

Statistics

What you will learn in Chapter 4

Why pictures are worth “a thousand words”

How to create:HistogramPolygonOther charts/graphs

Using SPSS to create & modify charts

Different types of charts and their uses

Why Illustrate Data?

When describing a set of scores you will want to use two things…One score for describing the group of data

Measure of Central TendencyMeasure of how diverse or different the

scores are from one anotherMeasure of Variability

However, a visual representation of these two measures is much more effective when examining distributions.

Ten Ways to a Great Figure

Minimize the “junk”Plan before you start creatingSay what you mean…mean what you sayLabel everythingCommunicate ONE ideaKeep things balancedMaintain the scale in the graphRemember…simple is bestLimit the number of wordsThe chart alone should convey what you

want to say

Frequency Distributions

Method of tallying, and representing the number of times a certain score occursGroup scores into interval classes/ranges

Creating class intervalsRange of 2, 5, 10, or 20 data points10-20 data points cover the entire range of

dataLargest interval goes at the top

HistogramsClass Intervals Along the x-Axis

HistogramsHand Drawn Histogram

HistogramTally-Ho Method

Frequency PolygonA “continuous line that represents the

frequencies of scores within a class interval”

Cumulative Frequency Distribution

Fat & Skinny of Frequency Distributions

Distributions can be different in four different ways…Average valueVariabilitySkewnessKurtosis

Average Value

Variability

SkewnessPositive & Negative Skewness

KurtosisPlatykurtic & Leptokurtic

Cool Ways to Chart DataColumn Chart

Cool Ways to Chart DataLine Chart

Cool Ways to Chart DataPie Chart

Using the Computer to Illustrate Data

Creating Histogram Graphs

Using the Computer to Illustrate Data

Creating Bar Graphs

Using the Computer to Illustrate Data

Creating Line Graphs

Using the Computer to Illustrate Data

Creating Pie Graphs

Glossary Terms to Know

Frequency distributionClass intervalHistogramFrequency PolygonCumulative Frequency Distribution

Ogive SkewnessKurtosis

PlatykurticLeptokurtic