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Descriptive Statistics

SpEd 642

Brownbridge"

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Key Terms"n  Raw score"n  Nominal, ordinal,

interval and ratio scales"

n  Derived scores"n  Standard scores"n  Descriptive statistics"n  Measures of central

tendency"n  Normal distribution "n  Frequency distribution"n  Mode, median, and

mean"

n  Bimodal distribution"n  Multimodal

distribution"n  Frequency polygon"n  Standard deviation"n  Variability"n  Range"n  Skewed data"n  Positively skewed"n  Negatively skewed"n  Percentile ranks"

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Why Is Measurement Important?"

n  When students are evaluated for special education, the information from the evaluation will be used to make important, life-changing decisions. "

n  So the data from assessment must be collected in an objective and accurate way. "

n  If professionals rely on only subjective data or information collected in a non-standard ways, then the results may show bias and may not be useful in determining the student’s actual status (e.g., average, below average, above average) in relation to peers or standards for grade level."

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Accurate interpretation of test results requires understanding of

measurement and what it means.

Statistics are used to interpret data from standardized assessments.

Statistics show how a particular score

relates to what is typical.!

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Final Examination Raw Scores"

89 94 69 82 83 98 62 98 89 92 91 89 89 90 92 78

The numbers to the left are raw scores from 16 students who took a final examination. Raw scores represent the number of items correct (sometimes called the obtained score).

What conclusions could you draw from these raw scores?

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What Makes Numerical Data Meaningful?"

n  By themselves, raw scores do not mean much. For these scores to have meaning, we have to place them in a context or a frame of reference."

n  We need to know, for example, how many items were on the test itself. Then we can determine what percentage of correct items each student""achieved."

n  Other “frames of reference” might include: comparisons to each student’s previous achievement, comparisons with students taking a similar course, or comparisons to a large norm sample."

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Numerical Scales"

n  The scale on which we arrange numbers can be another frame of reference for gaining meaning from them. "

n  Depending on what scale we use, numbers can denote different meanings. "

n  The four basic numerical scales include: nominal, ordinal, interval and ratio scales."

1 2 3 4 5 6 7 8 9 10

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Types of Numerical Scales"

n  nominal scale: numbers used like names; a scale of measurement in which there is no inherent relationship among adjacent values; numbers used for the purpose of identification; no values are assigned to categories; impossible to add, subtract, multiply or divide"

n  ordinal scale: scales on which values of measurement are ordered from best to worst or from worst to best; on ordinal scales, the difference between adjacent values is unknown and variable; rank order; numbers indicate greater or lesser, not equidistant units

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Types of Numerical Scales"

n  interval scale: numbers used for identification that rank greater or lesser quality or amount and are equidistant; no absolute zero"

n  ratio scale: scales of measurement in which the difference between adjacent values is equal and in which there is logical and absolute zero; allows for direct comparisons"

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Questions to ask about numerical scales…"

n  Do adjacent items have a mathematical relationship to each other?"

"

n  Are the values in the scale equidistant?""

n  Can you “average” the values?""

n  Is there an absolute zero? At zero is there a complete absence of the value (e.g., light)?

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What kind of scale is each of these?

n  highest to lowest examination scores"

n  IQ scores"n  mile markers"n  Kelvin temperature

scale"n  chocolate, strawberry,

tutti frutti, mocha"

n  class rank"n  brown, blue, green,

gray"n  weight in pounds"n  good, better, best"n  age equivalents"n  percentiles"n  heaviest to lightest in

weight"

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Problem: What Difference Does the "Numerical Scale Make?"

On the Wechsler Intelligence Scale for Children - IV (WISC-IV), Harry earns an IQ of 52, and Ralph earns an IQ of 104.""Their teacher concludes that Ralph is twice as smart as Harry."""

Why is this conclusion incorrect?!

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Looking for Differences"

n  When students are evaluated for special education, we are interested in knowing what the student’s academic strengths and weaknesses may be, and we need to find out if the student is performing differently from what is expected of students of the same age or at the same grade level. "

n  Students who perform at a much lower level than others may have a disability and be in need of special help."

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Making Comparisons"

n  When we assess students for possible identification for special education, we need to compare the scores of the target students with those achieved by other students of the same age or grade level. In order to make these comparisons, we use descriptive statistics. "

n  Descriptive statistics are simply standard ways of describing sets of data."

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Derived Scores"

n  Derived scores is a general term for raw scores that are transformed (using expectancy tables) into developmental scores (age or grade equivalent) or to scores of relative standing (rank). Examples of derived scores include: "n  percentile ranks, standard scores, grade equivalents,

age equivalents, or language equivalents."n  By converting raw scores to derived scores, we can

compare one student’s score to those obtained by a large group of students of the same age or grade level. "

n  We can determine, for example, whether one student’s score is average, below average or above average when compared to others who have taken the same test."

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Types of Derived Scores"

"Grade Equivalents—a score that translates test performance into an estimated grade level; expressed in grades and tenths of grades (uses ordinal data)"

""Grade equivalent scores do not describe the student’s current instruction level; they are not diagnostic. They are not even indicative of which test questions were answered correctly. Grade equivalents simply indicate that the student got the same number right as is typical for students at that grade level. Two students, for example, could earn the same grade score but not answer any of the same questions correctly."

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Other Types of Derived Scores"

n  Age Equivalents—a score that translates test performance into an estimated age; reported in years and months (uses ordinal data)"

n  Percentiles—a measure of relative standing; translates test performance into the percentage of the norm group that performed as well as or poorer than the student on the same test (ordinal data)."

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Percentiles"n  Using percentile ranks is a method of ranking each

score in a set of data along the continuum of the normal distribution."

n  The extreme scores are ranked at the top and bottom, and very few people obtain scores at the extreme ends."

n  Percentiles range from the 99.9th percentile to less than the 1st percentile."n  A person who scores around the average, say 100 on an

IQ test, would be ranked in the middle at the 50th percentile."

n  A person who scores in the top fourth would be above the 75th percentile; in other words the student scored as well as or better than 75% of the students in that particular age group or grade level."

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Standard Scores"

n  Standard score is the general name for derived scores that have been transformed to produce a distribution with a predetermined mean and Standard Deviation (e.g., IQ scores, z-scores, t-scores)."

n  Standard scores are based on an interval scale with equal units."

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Examples of Standard Scores"

n  IQ scores—standard scores with a mean of 100 and a standard deviation of 15."

n  Z-scores—standards scores with a mean of 0 and a standard deviation of 1"

n  T-scores—standard scores with a mean of 50 and a standard deviation of 10"

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Classification of Standard Scores"

Standard Score""Less than 70"70-79"80-89"90-109"110-119"120-129"130 and higher"

Classification""Developmentally delayed"Borderline"Low Average"Average"High Average"Superior"Very Superior"

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Case Study: Sally"

DIRECTIONS: Compare Sally’s performance on three achievement measures. Draft an explanation you would offer to Sally’s parents to help them understand the differences in Sally’s performance in math, spelling and reading. Also explain the difference between standard scores and percentile ranks."

TEST DATA""

Math: " "Standard Score " "97"" " "Percentile Rank " "45"

"Spelling:" "Standard Score " "85"" " "Percentile Rank " "18"

"Reading:" "Standard Score " 115"" " "Percentile Rank " "81"

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Measures of Central Tendency"

The three measures of central tendency are: "mode, median and mean. These statistics are different ways of describing how scores relate to average."

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Mode"In a distribution (group of scores), the mode is the"most frequently obtained score. Sometimes a"distribution has more than one mode."

n  bimodal distribution: a distribution that has two modes"

n  multimodal distribution: a distribution with three or more modes "

A frequency polygon is a graph of showing how"often each score occurs in a set of data. For

example"with a bimodal distribution, the frequency"polygon will show two “peaks” where the most"frequently occurring scores appear. ""

" " " "(See Activity 3.4, pp. 105.06)"

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Median"

n  The median is a score that divides the top 50 percent of test takers from the bottom 50 percent; it is the point in a distribution, above which 50 of the cases (not the scores) occur and below which 50 of the cases occur. "

To calculate the median, arrange the scores in rank order. Each score must be listed each time it occurs. When the list is complete, count halfway down the list of scores. In a set of data with an odd number of scores, the median will an actual score that is middlemost in the distribution."

75, 75, 74, 73, 71 Median = 74. "n  In a set of data with an even number of scores, the median

is the middle-most score even though that score does not actually exist in that set of data."

36, 35, 33, 30 Median = 34!

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Mean"

n The mean is the arithmetic average of scores in a distribution; it is the statistic that best measures average performance."

n  To calculate the mean, add up all the scores in the distribution and divide by the number of scores. "

"

88, 84, 81, 78, 72"Sum = 403"

403/5 = 80.6"Mean = 80.6"

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Distributions"n  One way to look at a set of data is to rank

the scores from highest to lowest and then count to see how often a particular score occurred."n  distribution: a collection of scores"n  frequency distribution: a count of how many

times a score appears in a distribution"n  normal distribution: symmetrical distribution

with a single numerical representation for mean, median and mode; a normal distribution hypothetically represents the way test scores would fall if a particular test were given to every single student of the same age or grade level."

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Normal Distribution"The distribution of IQ scores is an example of a normal distribution for which the mode, median and mean are all the same number."On tests measuring achievement or intelligence, the mean is 100. One hundred is also the middle-most score (median) and the most frequently occurring score (mode). "

Standard deviations

Percentile ranks

z scores

Tscores

Wechsler lOs (SO= 15)

Wechsler subtest scores

(SO= 3)

Stanford-Binet lOs (SO= 16)

· ·-···""'

Stanford-Binet subtest scores

(SO= 8)

Stanine

(f) Q) (f)

Cll u

0 '-Q)

.D E ::J z

--40 -30 -20

--4 -3 -2

10 20 30

55 70

4

52 68

18 26 34

4%

13.59% 34.13% 34.13% 13.59%

-10 Mean Test Score

+10

5 10 20 30 50 70 80 90 95

-1 0 +1

40 50 60

85 100 115

7 10 13

84 100 116

42 50 58

7% 12% 17% 20% 17% 12% 7%

2 3 4 5 6 7 8

4-1. Relationship of the normal curve to various types of standard scores.

+20 +30 +40

99

+2 +3 +4

70 80 90

130 145

16 19

132 148

64 72 80

4%

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Usefulness of Measures of Central Tendency"

n  Knowing what is average or typical aids educators in monitoring progress and determining which students are performing well above or well below expectations for their age or grade level. "

n  Finding the “outlyers,” the students who perform very differently from average, can assist educators in tailoring instruction or finding supports to meet the needs of these very different students.!

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Practice Data Set"

n  For this data set, provide the following: frequency distribution, mean, mode, median and range.!

!84 "97 "86"72 "58 "78"96 "95 "85"78 "60 "79"84 "92 "85"89 "70 "80"99 "87 "74"85 "74 "81"74 "86 "84"63 "78 "82"

""Mean: " "Mode: " "Median:" "Range:

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Measures of Difference"

n  For the purposes of identifying students who need special education, educators must look for scores that show performances that are much lower than average."

n  To find these differences, educators use statistics that are called measures of dispersion. Measures of dispersion are used to calculate how scores spread from the mean."

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Which of the following statements is true about the two sets of scores X and Y?"

"X "= "3 "4 "5 "6 "7 "8 "9""Y "= "1 "3 "5 "6 "7 "9 12 """"a. Set X and set Y are equal""b. "set X and set Y are normal "distributions""c. "set X is more variable than set Y""d. "set Y is more variable than set X"

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Measures of Dispersion"

variability: describes how scores vary from average""range: the amount of difference between the

highest score and the lowest score (e.g., subtract the lowest score from the highest score)"

"variance: a numerical index describing the

dispersion of a set of scores around the mean of the "distribution; describes the total amount that a group of scores varies in a set of data (e.g., the average of the sum of squares)

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Standard Deviation"

n  The standard deviation (SD) is a measure of dispersion that describes the average degree of difference of a score from the mean."

n  The standard deviation is the square root of the variance."

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Steps to Computing Standard Deviation"

n  Arrange the scores in a frequency distribution."

n  Calculate differences between mean and each score in the distribution."

n  Square the differences."n  Sum up the squares (e.g., sum of squares)."n  Divide the sum of squares by the number of

scores. This number equals the variance."n  Take the square root of the variance. This

number is the standard deviation."

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Practice Problem:Finding Central Tendency and Variance"

Ms. Robbins administers a test to ten!children in her class. The children earn the!following scores: 14, 28, 49, 49, 49, 77, 84,!84, 91, and 105. For this distribution of!scores, find the following: mode, median,!mean, range, variance and standard!deviation.!

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89 94 69 82 83 98 62 98 89 92 91 89 89 90 92 78

For this distribution of scores, find the following: FREQUENCY DISTRIBUTION: FREQUENCY POLYGON: RANGE: MEAN: MEDIAN: MODE: STANDARD DEVIATION:

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Term Scores for Students in Educational Foundations"

401 436 418 415 440 360 447 428 431 413 391 379 423 438 396 377

For this distribution of scores, find the following: FREQUENCY DISTRIBUTION: RANGE: MEDIAN: MODE: MEAN: STANDARD DEVIATION:

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Drawing Conclusions"

The following statements about Test A and Test B are true:""Tests A and B have means of 100""Test A has a standard deviation of 15""Test B has a standard deviation of 5"

"Following classroom instruction, the pupils in Mr. Radley’s room earn

an"average score of 130 on Test A. Pupils in Ms. Purple’s room earn an"average score of 110 on Test B. On this basis, the local principal"concludes that Mr. Radley’s students learn more than Ms. Purple’s.

What"is fallacious about this conclusion?""n  Assuming the the pupils were equal prior to instruction, what

conclusions could the principal legitimately draw concerning the test scores? Why?!

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Standard Deviation and the Normal Distribution"

In a normal distribution, more than 68% of"the scores fall within one standard "above or below the mean. Approximately"95% of the scores are found within two"standard deviations above and below the"mean. See Figure 3.4 on page 116."

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Percentages of PopulationThat Fall within SD Units"

SD Units""

X to 1 SD""

1 to 2 SD""

3 to 3 SD""

3 to 4 SD"

Percentage of Population""

34.13% (68.26%)""

13.59% ""

2.14% ""

.13% "

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Another Exercise"Use the normal distribution to answer the following questions. " "1.   What percentage of the scores would fall between the z

scores of -2.0 and +2.0? ____"2.   What percentile rank would be assigned to the z score of

0? _____"3.   What percentile rank would represent a person who

scored at the z score of +3.0? _____"4.   Approximately what percentile rank would be assigned

for the IQ score of 70? _____"5.   Approximately how many people would be expected to

fall in the IQ range represented by the z scores of +3.0 to +4.0? _____"

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Practice Exercise"

Using the normal distribution, answer the following questions."

1. Approximately what percentage of people would be expected to fall in the IQ range represented by the z- scores of 0 and +1?"

2. About what percentile rank would represent a person who scored at the z score of -2?"

3. What percentile rank would be assigned to Stanine 5?"

Convert the following z scores to T- scores: !Hint: T-Score = Mean + (SD)(z)"

4. "Sam’s z -score = 1.33"5. Sean’s z-score = 0"

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Mean Differences"

n  Test results should always be interpreted with caution."

n  Many tests that have been used historically to diagnose disabilities such as mental retardation have been shown to exhibit mean differences."

n  A specific cultural or linguistic group may have a different mean or average score than that reported for most of the population; this is a mean difference."

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Skewed Data"

n  skewed: an asymmetrical distribution; the distribution of scores below the mean is not a mirror image of the distribution above the mean"

n  positively skewed: an asymmetrical distribution in which scores tail off to the higher end of the continuum; a distribution in which there are more scores below the mean than above it"

n  negatively skewed: an asymmetric distribution in which scores tail off to the low end; a distribution in which there are more scores above the mean than below it"

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What would be the"implications for teaching when"the distribution of scores on a"teacher-made test resulted in a"negatively skewed distribution?"

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Questions"

n  Why is it important for teachers to understand measures of central tendency?"

"n  What are the major differences between

standard scores and percentile ranks?"

n  How are national norms and local norms useful? Under what circumstances?"