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Educational Research Chapter 11 Descriptive Statistics Gay, Mills, and Airasian.
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Transcript of Educational Research Chapter 11 Descriptive Statistics Gay, Mills, and Airasian.
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)(