Educational Research

Chapter 12Descriptive Statistics

Gay, Mills, and Airasian10th Edition

Topics Discussed in this Chapter

Preparing data for analysis Types of descriptive statistics

Central tendency Variation Relative position Relationships

Calculating descriptive statistics

Preparing Data for Analysis

Issues Scoring procedures Tabulation and coding Use of computers

Scoring Procedures Instructions

Standardized tests detail scoring instructions Teacher-made tests require the delineation of

scoring criteria and specific procedures Types of items

Selected response items - easily and objectively scored

Open-ended items - difficult to score objectively with a single number as the result

Tabulation and Coding Tabulation is organizing data

Identifying all information relevant to the analysis

Separating groups and individuals within groups Listing data in columns

Coding Assigning names to variables

EX1 for pretest scores SEX for gender EX2 for posttest scores

Tabulation and Coding Reliability

Concerns with scoring by hand and entering data

Machine scoring Advantages

Reliable scoring, tabulation, and analysis Disadvantages

Use of selected response items, answering on scantrons

Tabulation and Coding Coding

Assigning identification numbers to subjects

Assigning codes to the values of non-numerical or categorical variables

Gender: 1=Female and 2=Male Subjects: 1=English, 2=Math, 3=Science,

etc. Names: 001=John Adams, 002=Sally

Andrews, 003=Susan Bolton, … 256=John Zeringue

Computerized Analysis Need to learn how to calculate

descriptive statistics by hand Creates a conceptual base for

understanding the nature of each statistic Exemplifies the relationships among

statistical elements of various procedures Use of computerized software

SPSS-Windows Other software packages

Descriptive Statistics Purpose – to describe or

summarize data in a manner that is both understandable and short

Four types Central tendency Variability Relative position Relationships

Descriptive Statistics Graphing data – a

frequency polygon Vertical axis

represents the frequency with which a score occurs

Horizontal axis represents the scores themselves

SCORE

9.08.07.06.05.04.03.0

SCORE

Fre

qu

en

cy

5

4

3

2

1

0

Std. Dev = 1.63

Mean = 6.0

N = 16.00

Quiz 1 Results

Central Tendency Purpose – to represent the typical

score attained by subjects Three common measures

Mode Median Mean

Central Tendency Mode

The most frequently occurring score Appropriate for nominal data Look for the most frequent number

Median The score above and below which 50% of all

scores lie (i.e., the mid-point) Characteristics

Appropriate for ordinal scales Doesn’t take into account the value of all scores

Look for the middle # (if 2 are in contention, get the mean of these 2 numbers.

Central Tendency Mean

The arithmetic average of all scores Characteristics

Advantageous statistical properties Affected by outlying scores Most frequently used measure of central

tendency Add all of the scores together and

divide by the number of Ss

Calculate for the following data points: S1 = 10 S2 = 12 S3 = 14 S4 = 10 S5 = 14 S6 = 12 S7 = 12 S8 = 12

??= ???

Mode Median Mean

You know the central score, do you need anything else? What is the mean of the following:

10, 20, 200, 10, 20 What is the mean of the following:

51, 52, 53, 52, 52 Is there more we want to know

about the data than just what is the middle point?

Quiz 1: Central Tendency Count: 25 Average/ Mean: 79.7 Median: 83.5

Variability Purpose – to measure the extent to

which scores are spread apart Four measures

Range Variance Standard deviation (there are others, but these are the

only ones we are going to talk about)

Variability Range

The difference between the highest and lowest score in a data set

Characteristics Unstable measure of variability Rough, quick estimate

Calculate What is the range of the following:

10, 20, 200, 10, 20 What is the range of the following:

51, 52, 53, 52, 52

Quiz 1 Count: 25 Average: 79.7 Median: 83.5 Maximum: 93.4 Minimum: 0.0

Variability

Variance The average squared deviation of all

scores around the mean Characteristics

Many important statistical properties Difficult to interpret due to “squared” metric Used mostly to calculate standard deviation

Formula

Variance

10 - 52 = -42 20 - 52 = -32 200-52 = 148 10 - 52 = -42 20 - 52 = -32

51 - 52 = -1 52 - 52 = 0 53 - 52 = 1 52 - 52 = 0 52 - 52 = 0

Variance 10 - 52 = -422 =

1764 20 - 52 = -322 =

1024 200-52 = 1482

=21904 10 - 52 = -422 =

1764 20 - 52 = -322 =

1024

51 - 52 = -12 = 1 52 - 52 = 02 = 0 53 - 52 = 12 = 1 52 - 52 = 02 = 0 52 - 52 = 02 = 0

Variance 10 - 52 = -422 = 1764 20 - 52 = -322 = 1024 200-52 = 1482 =21904 10 - 52 = -422 = 1764 20 - 52 = -322 = 1024

27480

27480/5 = 5496Variance = 5496

51 - 52 = -12 = 1 52 - 52 = 02 = 0 53 - 52 = 12 = 1 52 - 52 = 02 = 0 52 - 52 = 02 = 0

2

2/5=.4Variance = .4

Variability Standard deviation

The square root of the variance Characteristics

Many important statistical properties Relationship to properties of the normal

curve Easily interpreted

Formula

Standard Deviation 10 - 52 = -422 = 1764 20 - 52 = -322 = 1024 200-52 = 1482 =21904 10 - 52 = -422 = 1764 20 - 52 = -322 = 1024

2748027480/5= 5496=

Variance ____√5496 = 74.13 = SD

51 - 52 = -12 = 1 52 - 52 = 02 = 0 53 - 52 = 12 = 1 52 - 52 = 02 = 0 52 - 52 = 02 = 0

2 2/5=.4; Variance = .4 __ √.4 = .63 = SD

So now you know middle # and spreadoutedness How can you use that information to

standardize all of the scores to have the same meaning.

First set of scores has a mean of 52 and a SD of .63; second set has a mean of 52 and a SD of 74.13. How do we compare an individual score on first to an individual score on second?

Quiz 1: Variance Count: 25 Average: 79.7 Median: 83.5 Maximum: 93.4 Minimum: 0.0 Standard Deviation: 18.44

The Normal Curve

A bell shaped curve reflecting the distribution of many variables of interest to educators

Gives a visual way of identifying where one person’s scores fit in with the rest of the people.

Normal Curve

The Normal Curve Characteristics

Fifty-percent of the scores fall above the mean and fifty-percent fall below the mean

The mean, median, and mode are the same values

Most participants score near the mean; the further a score is from the mean the fewer the number of participants who attained that score

Specific numbers or percentages of scores fall between ±1 SD, ±2 SD, etc.

The Normal Curve Properties

Proportions under the curve ±1 SD = 68% ±1.96 SD = 95% ±2.58 SD = 99%

Skewed Distributions None - even

Positive – many low scores and few high scores

Negative – few low scores and many high scores

Skewed Distribution Which direction are the following scores

skewed: 12,4,5,13,4,4,1,3,1,3,1,3,1,5

Step 1: Reorder from lowest to highest 1,1,1,3,3,3,4,4,4,5,5,12,13

Step 2: Graph these numbers Step 3: Compare the graph to the

pictures we showed above (tail goes toward the direction… tail to the right, positive; tail to the left, negative)

Skewed Distribution Example

1 3 41 3 4 51 3 4 5 12 13

Skewed Distribution Example

0

0.5

1

1.5

2

2.5

3

1 3 4 5 12 13

1st Qtr

Measures of Relative Position Purpose – indicates where a score

is in relation to all other scores in the distribution

Characteristics Clear estimates of relative positions Possible to compare students’

performances across two or more different tests provided the scores are based on the same group

Measures of Relative Position Types

Percentile ranks – the percentage of scores that fall at or above a given score

Standard scores – a derived score based on how far a raw score is from a reference point in terms of standard deviation units

z score T score Stanine

Measures of Relative Position z score

The deviation of a score from the mean in standard deviation units

Characteristics Mean = 0 Standard deviation = 1 Positive if the score is above the mean and

negative if it is below the mean Relationship with the area under the normal curve

Measures of Relative Position

T score – a transformation of a z score Characteristics

Mean = 50 Standard deviation = 10 No negative scores

Measures of Relative Position Stanine – a transformation of a z

score Characteristics

Nine groups with 1 the lowest and 9 the highest

Measures of Relationship: Correlations

Purpose – to provide an indication of the relationship between two variables

Characteristics of correlation coefficients Strength or magnitude – 0 to 1 Direction – positive (+) or negative (-)

Types of correlation coefficients – dependent on the scales of measurement of the variables

Spearman rho – ranked data Pearson r – interval or ratio data

Measures of Relationship

Interpretation – correlation does not mean causation

Formula see page 316 in your text book to discuss the formula for the Pearson r correlation coefficient.

Calculating Descriptive Statistics

Using SPSS Windows Means, standard deviations, and

standard scores The DESCRIPTIVE procedures

Correlations The CORRELATION procedure

Objectives 10.1, 10.2, 10.3, & 10.4