Educational Research

Chapter 11Descriptive Statistics

Gay, Mills, and Airasian

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

Objectives 1.1 & 1.2

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

Objectives 2.1, 2.2, & 2.3

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

Objectives 1.4 & 1.5

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

Objectives 2.2 & 2.3

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

Objective 2.4

Descriptive Statistics Purpose – to describe or

summarize data in a parsimonious manner

Four types Central tendency Variability Relative position Relationships

Objective 2.4

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

Objectives 3.1 & 3.2

Central Tendency Purpose – to represent the typical

score attained by subjects Three common measures

Mode Median Mean

Objective 4.1

Central Tendency Mode

The most frequently occurring score Appropriate for nominal data

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 each and

every score in the data

Objectives 4.2, 4.3, & 4.4

Central Tendency Mean

The arithmetic average of all scores Characteristics

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

tendency Formula

Objectives 4.2, 4.3, & 4.4

Variability Purpose – to measure the extent to

which scores are spread apart Four measures

Range Quartile deviation Variance Standard deviation

Objective 5.1

Variability Range

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

Characteristics Unstable measure of variability Rough, quick estimate

Objectives 5.2 & 5.3

Variability

Quartile deviation One-half the difference between the

upper and lower quartiles in a distribution

Characteristic - appropriate when the median is being used

Objectives 5.2 & 5.3

Variability

Variance The average squared deviation of all

scores around the mean Characteristics

Many important statistical properties Difficult to interpret due to “squared”

metric Formula

Objectives 5.2 & 5.3

Variability Standard deviation

The square root of the variance Characteristics

Many important statistical properties Relationship to properties of the normal

curve Easily interpreted

Formula

Objectives 5.2 & 5.3

The Normal Curve

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

See Figure 14.2 See the attached slide

Objective 6.1

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.

Objectives 6.1, 6.2, & 6.3

The Normal Curve Properties

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

Cumulative proportions and percentiles

Objectives 6.3 & 6.4

Skewed Distributions Positive – many low scores and few high

scores Negative – few low scores and many

high scores Relationships between the mean,

median, and mode Positively skewed – mode is lowest, median

is in the middle, and mean is highest Negatively skewed – mean is lowest, median

is in the middle, and mode is highest

Objectives 7.1 & 7.2

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

Objectives 7.1 & 7.2

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

Objectives 9.3 & 9.4

Measures of Relative Position z score

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

The basic standard score from which all other standard scores are calculated

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

Objective 9.5

Measures of Relative Position

z score (continued) Possible to calculate relative

standings like the percent better than a score, the percent falling between two scores, the percent falling between the mean and a score, etc.

Formula

Objective 9.5

Measures of Relative Position

T score – a transformation of a z score where T = 10(z) + 50 Characteristics

Mean = 50 Standard deviation = 10 No negative scores

Objective 9.6

Measures of Relative Position Stanine – a transformation of a z

score where the stanine = 2(z) + 5 rounded to the nearest whole number Characteristics

Nine groups with 1 the lowest and 9 the highest

Categorical interpretation Frequently used in norming tables

Objective 9.7

Measures of Relationship

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

Objectives 8.1, 8.2, & 8.3

Measures of Relationship

Interpretation – correlation does not mean causation

Formula for Pearson r

Objective 8.2

Calculating Descriptive Statistics Symbols used in statistical analysis General rules for calculating by hand

Make the columns required by the formula

Label the sum of each column Write the formula Write the arithmetic equivalent of the

problem Solve the arithmetic problem

Objectives 10.1, 10.2, 10.3, & 10.4

Calculating Descriptive Statistics

Using SPSS Windows Means, standard deviations, and

standard scores The DESCRIPTIVE procedures Interpreting output

Correlations The CORRELATION procedure Interpreting output

Objectives 10.1, 10.2, 10.3, & 10.4

Calculating Descriptive Statistics

See the Statistical Analysis of Data module on the web site for problems related to descriptive statistics

Formula for the Mean

n

xX

Formula for Variance

1

2

2

2

NN

xS

x

x

Formula for Standard Deviation

1

2

2

NN

xSD

x

Formula for Pearson Correlation

N

yy

N

xx

N

yxxy

r2

2

2

2

Formula for z Score

sxXx

Z)(