Chapter 5 Alternative ways to report test Scoress, 2015 ppt

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Chapter V:

Alternative

Ways To

Report Test

Scores

Rosa Padilla Castro de Casamayor

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2 Rosa Padilla Castro

INTRODUCTION

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3 Master Rosa Padilla

Definition of Statistics

• Descriptive statistical methods have their beginnings in the inventories kept by early civilizations, such as the Babylonians, Egyptians, and Chinese. For example, the Old Testament of the

Bible (Number 1, Luke 2: 1,2) refers to the numbering or counting of the people of Israel and to the casting of lots for selection by chance, and the Romans kept careful counts of people, possessions, and wealth in the territories they conquered. These early methods were primarily lists and counts kept for purposes of taxation and military conscription.

– Two main branches: • Descriptive statistics • Inferential statistics

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Master Rosa Padilla

The Branches of Statistics

Descriptive statistics consist of methods for organizing displaying and describing data

by using table, graphs and summary statistic.

Statistics

Descriptive Statistics

Inferential Statistics

Inferential statistics is a methods that making decisions

or predictions about a population based on sampled

data, i.e. the set of methods whose purpose is to infer or induce behavioral laws of a

population from a sample survey, which will help us in

decision-making under a certain degree of confidence, this confidence is measured by

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Why study statistics?

Students, like professional people, must be able to read and understand the various statistical studies performed in their field. To have understanding, they must be knowledgeable about the vocabulary, symbols, concepts and statistical procedures used in these studies.

Students and professional people may be called on to conduct research in their field, since statistical procedures are basic to research. To accomplish this, they must be able to design experiments; collect, organize, analyze, and summarize data; and possibly make reliable predictions or forecasts for future use. They must be able to communicate the results of the study in their own words.

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Definitions

A Population is the complete collections of measurement, objects, or individuals under study. Its size is usually denoted by “N"

ASample is some of portion or subset taken from a population. Its size is usually denoted by "n." If every member of a population is evaluated, a census has been performed, and any summary value of all of the individual measurements is called a parameter. If only a subset or sample of a population has been evaluated, any summary value of such measurements is called a statistic.

A Data numbers or measurements collected

A parameter A numerical measure that describes a variable (characteristic) of a population. e.g., the average height of all Rwandans

A statistic A numerical measure that describe a variable (characteristic) of a sample (part of population). e.g., the average height of a sample of Rwandans.

A variable variable A variable is an attribute of a person or an object that varies.

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Definitions

Sources of data:

You begin every statistical analysis by identifying the source of the data.

Among the important sources of data are published sources,

experiments, and surveys.

Published Sources

Data available in print or in electronic form,

including data found on internet website. Primary data sources are those

published by the individual or group that collected the data. Secondary

data sources are those compiled from primary sources.

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Classification of Variables

Variables can be classified in several ways. One method of classification refers to the type and amount of information contained in the data. Data are either categorical or numerical. Another method is to classify data by levels of measurement, giving either qualitative or quantitative variables

CATEGORICAL OR NUMERICAL

Categorical variables produce responses that belong to groups or categories. For example, responses to yes/no questions are categorical. “Did you ever visit Gorillas Mountains in Rwanda?” are limited to yes or no answers. Sometimes categorical variables include a range of choices. For example, consider a faculty evaluation where students are to respond to statements such as “The instructor in this course was an effective teacher” (1: strongly disagree; 2: slightly disagree; 3: neither agree nor disagree; 4: slightly

agree; 5: strongly agree)

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Types of Variables

• _{A quantitative}

variable has no gaps

between its values.

All values or

fractions of values

have meaning. Age

is an example of

quantitative

variable.

• _{A qualitative}

variable has gaps

between its values.

For example,

gender is a

qualitative

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Master Rosa Padilla

LEVEL OF MEASUREMENT



Nominal-Level

data is merely descriptive (e.g. religion, country name, sex). Any

assigned numerical value is merely for convenience (e.g. Catholic = 1, Adventist = 2,

Other = 3)



Ordinal-Level

data has rank order, though intervals between data points cannot be

considered equal (e.g. Income (high/medium/low); Severity (poor, average, high).



Interval-Level

This kind of measurement not only assigns rank or order but points

out the relative qualitative as well as quantitative difference. The major strength of this

scale lies in the fact that they have equal units of measurement. However they do not

possess a true zero. Example: Fahrenheit or centigrade scale here the zero does not

indicate the absence of heat.

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Types of Variables(Experimental design)

• _{Independent variable}

Variable controlled by the researcher; changes in

this variable may produce changes in the

dependent variable

• _{Dependent variable}

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A. Measures of central tendency

1. Mean 2. Median 3. Mode

B. Measures of variability

1. Normal bell-shaped curve or distribution 2. Range

3. Standard deviation 4. Skewed data/scores

C. Statistical techniques (to express data)

1. Frequency plot 2. Normative data 3. Percentiles 4. Correlations

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Nominal

Ordinal

Scale

Definition

Unordered

Ordered

Metric-numeric values

Examples

Gender,

marital status

Satisfaction,

age range

Age, weight, height,

income

Measure of

central

tendency

Mode

Median, mode

Mean, median and mode

Measure of

dispersion

Min/max/Range/ ICR

Min/max/range/ICR/std.

deviation

Graphics

Pie/Bars

Bars

Histogram, line, scatter,

boxplot, stem-and-leaf,

normality plots, etc.

Procedures

Frecuencies /%

Frecuencies/descriptive

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Statistics

Central Tendency

They are computed to give a “center” around which the measurements in the data are distributed; ie. measures of central tendency provide information about the average or typical score in a data set:

Mean, median and mode

Measures of Variability

Indicate the degree of concentration data with respect to mean or how far away the measurements are from the center:

Variance, standard deviation, coefficient of variation, range, maximum and minimum

Point Measures (quantiles)

Divide an ordered set of data into groups with the same number of individuals, describe a participant’s performance compared to the performance of all other participants :

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Master Rosa Padilla

The Arithmetic Mean: The average score of a distribution of scores Measures of central tendency

n

x





The Mean is a measure of central value. What most people mean by “average” The arithmetic mean is calculated by summing all of the scores in a distribution and dividing by the number of scores. In statistical notation we can express this mean as follows:

.

29 .

89

7

625

7

40

70

100

150

140

80

45 cm

n

x











Interpretation: Inches average is 89.29 cm

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Median: Is the middle score in a set of ranked scores. The scores that divides the distribution into halves. It is sometimes called the counting average.

Measures of central tendency

Steps to computing the median

1. Line up scores from lowest to

highest

2. Count up to middle score

• _{If there is 1 middle score, that’s the}

median

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

_{Quartiles: Divide the data into 4 equal parts}



_{Deciles: Divide the data into 10 equal parts}



_{Percentiles: Divide the information into 100 equal parts}

Defines the order quantile as a variable value below which is a cumulative frequency.

Special cases are the percentiles, quartiles, and deciles.

Point Measures: Quantile

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Q₁: N/4 or the 25% of distribution

Q₂: N/2 or the 50% of distribution ( this is the same as the median of the distribution)

Q₃: 3N/4 or the 75% of distribution

D₁: N/10 or the 10% of the distribution D₂: N/20 or the 20% of the distribution D….

D₉: N/90 or the 90% of the distribution Deciles

P₁: N/1 or the 1% of the distribution P₁₀: N/10 or the 10% of the distribution P₂₅: N/25 or the 25% of the distribution P₅₀: N/50 or the 50% of the distribution P₇₅: N/75 or the 75% of the distribution P₉₀: N/90 or the 90% of the distribution P₉₉: N/99 or the 99% of the distribution Percentiles

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Examples (if we were analyzing the variable weight in newborns and the variable cholesterol in adults)

5% of newborns is extremely underweight. What weight is considered "too low"?

5 percentile

What weight is exceeded only by 25% of individuals? 75 percentile

Cholesterol is distributed symmetrically in the population. It is considered pathological extreme values. 90% of normal individuals between what values are normal individuals?

Between 5 and 95 percentile

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Measures of variability

1 )

(

2 2 2







n

x

n

x

s

1 )

(

2 2 2







f

x

n

f

x

s

In population:

_N

N

x











2 2

2

(

)

 _Variance:

Describes the total amount that a set of scores varies from the mean.

 _{Measure the degree of dispersion (variability) of the data, regardless of}

cause.

Groupuped data

In a sample:

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Standard Deviation:

Shows the data scatter about the mean.

A small standard deviation means that the group has small variability or relatively homogeneous.

At a distance of one half standard deviation of 68% will observations. At a distance of two half standard deviation of 95% will observations.

2 s

sd



So…what is a

standard deviation

?

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Commonly Reported Test Scores Based on the Normal Curve

A “bell-shaped” curve in which most of the scores are clustered around the mean; the farther from the mean, the less frequently the score occurs. distribution—a distribution characterized by a bell-shaped curve, and the mean = median = mode

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• Derived scores

are used to specify where the individual score falls on the

curve and how far above or below the mean the score falls.

The Normal Curve

• All scoring scales are drawn parallel to the baseline of the normal curve; and use the deviation from the mean as the reference to compare an individual score with the mean score of a group

• Raw scores are transformed into percentiles, stanine or other standard scores

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Percentiles and Percentile Ranks

Example: What percentile rank tells us is within this group what

position a children score’s represents,(the score is evaluate between

0-100)

75 points for example ?

And gives us some idea if this is high or low relative to group

Percentile

- is a measure that tells us what percent of the total

frequency scored at or below that measure.

Examples - 60th percentile.

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• _{Percentile ranks, like most of the standardized scores}

discussed, are relative placement scores. They show were a

student fits in with a larger norm group. However, unlike the

other standardized score scales discussed in this chapter,

percentile ranks are not very useful for comparing students

with each other because they typically involve unequal

intervals.

• _{A student}

’

s percentile rank on a test indicates the percentage

of students who scored lower in the comparison group.

For example, if a student is ranked in the 55

th

_{percentile, the}

student

’

s score was 55% better than the comparison group

who took the test.

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x f % C% x f % C% 6 1 5 5 6 2 8 8 7 0 5 7 3 12 19 8 2 10 15 8 4 15 35 9 2 10 25 9 6 23 58 10 2 10 35 10 4 15 73 11 3 15 50 11 2 8 81 12 4 20 70 12 1 4 85 13 3 15 85 13 2 8 92 14 2 10 95 14 1 4 96 15 1 5 100 15 1 4 100

20 100 26 100

The following tables talk about the scores that the students obtains from the last mid examination on Inferential Statistic and Math Level respectively

Scores from Inferential Statistic Scores from Math Level

Interpretation: In the course Inferential Statistic the 50% of the students got score less than 11, while that in Math the 58% is less than 9 of score

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Example

You have administered a test to a large group of students and the frequency distribution approximates the normal curve. The mean on this test is 70 and the standard deviation is 5. You are interested in four students: Francine, Albertine, Samuel and Pheneas. Their scores and percentile ranks are listed in the following table:

Student Test score Percentile rank Francine 70 50

albertine 75 84 Samuel 80 98 Pheneas 85 99

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Z

Scores

By itself, a raw score or X value provides very little information

about how that particular score compares with other values in the

distribution.

A score of X = 52, for example, may be a relatively low score, or an

average score, or an extremely high score depending on the mean

and standard deviation for the distribution from which the score was

obtained.

If the raw score is transformed into a z-score, however, the value of

the z-score tells exactly where the score is located relative to all the

other scores in the distribution.

The process of changing an X value into a z-score involves creating

a signed number, called a

z-score

, such that:

a.

The sign of the z-score (+ or –) identifies whether the X value is

located above the mean (positive) or below the mean (negative).

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Z

Scores

• _{When values in a distribution are converted to}

Z

scores, the distribution will have

–

_{Mean of 0}

–

_{Standard deviation of 1}

And it is called Normal Standard Distribution

• _Useful

–

_{Allows variables to be compared to one another}

even when they are measured on different scales,

have very different distributions, etc.

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• _{To compute a}

_Z

_score,

subtract the mean

from a raw score and

divide by the SD

• _{To convert a}

_Z

_score

back to a raw score,

multiply the

Z

score by

the SD and then add

the mean

SD

M

X

Z



(



)

M

SD

Z

X



(

)(

)



Transforming back and forth between X and z

Where:

X

is the raw score

M

is the mean

score

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A measure of an observation’s distance from

the mean.

The distance is measured in standard deviation

units.

If a z-score is zero, it’s on the mean.

If a z-score is positive, it’s above the

mean.

If a z-score is negative, it’s below the

mean.

If a z-score is 1, it’s 1 SD above the mean.

If a z-score is –2, it’s 2 SDs below the

mean.

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Examples of computing z-scores

5

3

2

1

6

3

2

1.5

5

10 -5

4 -1.25

6

3

4 .75

4

8 -4

2 -2

X

_X

X



X

SD

X

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Computing raw scores from z scores

1

2

3

5 -2

2 -4

2 -2

.5

4

2

10

12 -1

5 -5

10

5 SD

X

z





SD

zSD

X

ZSD

X

or

z

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Formula for converting a raw score to a z-score is:

SD

M

X

Z





Where:

X is the raw score M is the mean score SD is the standard deviation

Example: Let’s say an individual takes a statistic exam, the data is following:

Step:

1.Find Mean =

2. Variance= 3.Standard deviation= 4.Z-score= Studen t Raw Scor e (x)

(x)2

Z-score

1 15 225 0.66 2 10 100 -0.71 3 17 289 1.21 4 13 169 0.11 5 8 64 -1.26

63 847

6 .

12

5

63 _





n

x

1 ) ( , 1 )

( 2 ₂ 2 2

2      



n x n x s or n x x s

65 .

3

3 .

13 



SD

3 .

13

4 )

6 .

12 *

5

847 (

1 )

(

2 2

2

_



_

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Interpreting

Studen t Raw Score (x) Z-score Probabilistic

Normal value Percentile

1 15 0.66 0.7454

_74.54

%

2 10 -0.71

(1-0.7611)=0.2389

23.89 %

3 17 1.21 0.8869

_88.69

%

4 13 0.11 0.5438

_54.38

%

5 8 -1.26

(1-0.8962)=0.1038

10.38 %

From this example we can see the student “1” that individual

who scored a 15 on the exam has a z-score of 0.66. By

examining probabilistic Normal table you can see that this

student has a value is 0.7454, that mean has a percentile

score of approximately 74.54%

See if you can determine what the z-score would be for an individual who had a raw score of 13 on this same test, and determine the

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Suppose our basketball coach wants to estimate how many entering freshmen will be over 6’6” (78 inches) tall. Suppose the mean height of entering freshmen is 68 inches and the SD of height is 6.67 inches and there will be 1,000 entering

freshmen. How many are expected to be bigger than 78 inches?

Application

50 . 1 499 . 1 67 . 6 68 78       SD M X Z 1000*.0668= 66.8 Then the coach expected 66.8 = 67 freshmen

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1. For this set of sample scores, compute the mean, median, the variance and the standard deviation, and interpret the result.

3. If Judy got a z score of 1.5 on an in-class exam, what can we say about her score relative to others who took the exam?

a. it is above average b. it is average

c. it is below average d. it is a ‘B’

4. If a raw score is 8, the mean is 10 and the standard deviation is 4, what is the z-score?

a. -1.0 b. -0.5 c. 0.5 d. 2.0

SELF ASSESSMENT EXERCISE

15 10 17 12 14 8

5. If a distribution is normally

distributed, about what percent of the scores fall below +1 SD?

a. 15 b. 50 c. 84 d. 99

2. Suppose our basketball coach wants to estimate how many entering

freshmen will be over 6’6” (78 inches) tall. Suppose the mean height of

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