educational research chapter 12 descriptive statistics gay, mills, and airasian 10 th edition
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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