STATISTICAL TOOLSNEEDED

IN ANALYZING TEST RESULTS

Prof. Yonardo Agustin Gabuyo

Branches of StatisticsDescriptive statistics methods concerned w/ collecting, describing, and analyzing a set of data without drawing conclusions (or inferences) about a large group

Inferential statistics methods concerned with the analysis of a subset of data leading to predictions or inferences about the entire set of data or population.

Examples of Descriptive Statistics Presenting the Philippine population by constructing a graph indicating the total number of Filipinos counted during the last census by age group and sex

The Department of Social Welfare and Development (DSWD) cited statistics showing an increase in the number of child abuse cases during the past five years.

Examples of Inferential Statistics Source: Pilot Training Course on Teaching Basic Statistics by Statistical Research and Training Center Philippine Statistical

Association , Inc.

A new milk formulation designed to improve the psychomotor development of infants was tested on randomly selected infants. Based on the results, it was concluded that the new milk formulation is effective in improving the psychomotor development of infants.

Example

Teacher Ron-nick gave a personality test measuring shyness to 25,000 students. What is the average degree of shyness and what is the degree to which the students differ in shyness are the concerns of _________ statistics.

A. inferential B. graphic

C. correlational D. descriptive

ExampleThis is a type of statistics that give/s information about the sample being studied.

a. Inferential and co-relationalb. Inferentialc. Descriptived. Co relational

Inferential StatisticsSource: Pilot Training Course on Teaching Basic Statistics by Statistical Research and Training Center Philippine Statistical Association , Inc.

Larger Set(N units/observations) Smaller Set

(n units/observations)

Inferences and Generalizations

VARIABLES

Qualitative Quantitative

ContinuousDiscrete

Types of Variables

Qualitative variables variables that can be express in terms of properties, characteristics, or classification(non-numerical values).

Levels of Measurement1. Nominal

Numbers or symbols used to classify

2. Ordinal scale Accounts for order; no indication

of distance between positions3. Interval scale

Equal intervals; no absolute zero4. Ratio scale

Has absolute zero

Methods of Collecting Data Objective Method

Subjective Method

Use of Existing Records

Methods of Presenting Data Textual

Tabular

Graphical

Mean

Median

Mode

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of Variation

Range

Location

Maximum Minimum

Central Tendency

Percentile

Quartile Decile

Inter-quartile Range

Skewness

Kurtosis

Measures of Location

A Measure of Location summarizes a data set by giving a “typical value” within the range of the data values that describes its location relative to entire data set.Some Common Measures:Minimum, MaximumCentral TendencyPercentiles, Deciles, Quartiles

Maximum and Minimum Minimum is the smallest value in the

data set, denoted as MIN.

Maximum is the largest value in the data set, denoted as MAX.

Measure of Central Tendency A single value that is used to identify the “center” of the data it is thought of as a typical value of the distribution

precise yet simplemost representative value of the data

Mean Most common measure of the center Also known as arithmetic average

Sample Mean

Population Mean

Properties of the Mean may not be an actual

observation in the data set. can be applied in at least

interval level. easy to compute. every observation

contributes to the value of the mean.

Properties of the Mean subgroup means can be

combined to come up with a group mean

easily affected by extreme values

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Mean = 5 Mean = 6

Median Divides the observations into two

equal parts. If n is odd, the median is the

middle number. If n is even, the median is the

average of the 2 middle numbers. Sample median denoted as

while population median is denoted as

x~ ~

Properties of a Median may not be an actual observation in

the data set can be applied in at least ordinal level a positional measure; not affected by

extreme values

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median = 5

Mode the score/s that occurs most

frequently nominal average computation of the mode for

ungrouped or raw data

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 9

0 1 2 3 4 5 6

No Mode

Properties of a Mode can be used for qualitative as

well as quantitative data may not be unique not affected by extreme values may not exist

Mean, Median & ModeUse the mean when:

sampling stability is desired other measures are to be

computed

Mean, Median & Mode

Use the median when:

the exact midpoint of the distribution is desired

there are extreme observations

Mean, Median & Mode

Use the mode when:

when the "typical" value is desired

when the dataset is measured on a nominal scale

Example

Which measure(s) of central tendency is(are) most appropriate when the score distribution is skewed?

A. ModeB. Mean and modeC. Median D. Mean

Example

In one hundred-item test, what does Jay-R’s score of 70 mean?

A. He surpassed 70 of his classmate in terms of score

B. He surpassed 30 of his classmates in terms of score

C. He got a score above meanD. He got 70 items correct

Example

Which of the following measures is more affected by an extreme score?

A. Semi- inter quartile rangeB. Median C. Mode D. Mean

Example

The sum of all the scores in a distribution always equals

a. The mean times the interval size

b. The mean divided by the interval size

c. The mean times Nd. The mean divided by N 

Example

Teacher B is researching on family income distribution which is symmetrical. Which measure/s of central tendency will be most informative and appropriate?

A. ModeB. MeanC. MedianD. Mean and Median

Example

What measure/s of central tendency does the number 16 represent in the following score distribution? 14,15,17,16,19,20,16,14,16?

a.Mode onlyb.Mode and medianc. Median onlyd. Mean and mode

Example

What is the mean of this score distribution: 40, 42, 45, 48, 50, 52, 54, 68?

a. 51.88b. 50.88c. 49.88d. 68

Example

Which is the correct about MEDIAN?

a. It is measure of variabilityb. It is the most stable measure of central tendency

c. It is the 50th percentiled. It is significantly affected by extreme values

Example

Which measure(s) of central tendency can be determined by mere inspection?

a. Medianb. Modec. Meand. Mode and Median 

Example

Here is a score distribution: 98,93,93,93,90,88,88,85,85,85,86, 70,70,51,34,34,34,, 20,18,15,12,9,8,3,1.

Which is a characteristics of the scores distribution?

A. Bi-modal B. Tri-modalC. Skewed to the right D. No discernible pattern

Example

Which is true of a bimodal score distribution?

a. the group tested has two identical scores that appeared most.

b. the scores are either high or low.

c. the scores are high.d. the scores are low.

ExampleSTUDY THE TABLE THEN ANSWER THE

QUESTION:

Scores Percent of Students

0-59 2%60-69 8%70-79 39%80-89 38%90-100 13%

In which scores interval is the median?

a. In the interval 80 to 89b. In between the intervals of 60-69 and 70-79

c. In the interval 70-79d. In the interval 60-69

How many percent of the students got a score below 70?

a. 2%b. 8%c. 10%d. 39%

Percentiles Numerical measures that

give the relative position of a data value relative to the entire data set.

Percentage of the students in the reference group who fall below student’s raw score.

Divides the scores in the distribution into 100 equal parts (raw data arranged in increasing or decreasing order of magnitude).

The jth percentile, denoted as Pj, is the data value in the data set that separates the bottom j% of the data from the top (100-j)%.

EXAMPLESuppose JM was told that relative to the other scores on a certain test, his score was the 97th percentile. This means that 97% of those who took the test had scores less than JM’s score, while 3% had scores higher than JM’s.

DecilesDivides the scores in the

distribution into ten equal parts, each part having ten percent of the distribution of the data values below the indicated decile.

The 1st decile is the 10th percentile; the 2nd decile is the 20th percentile…..

9th decile is the 90th percentile.

Quartiles Divides the scores in the distribution

into four equal parts, each part having 25% of the scores in the distribution of the data values below the indicated quartile.

The 1st quartile is the 25th percentile; the 2nd quartile is the 50th percentile, also the median and the 3rd quartile is the 75th percentile.

Example

Robert Joseph’s raw score in the mathematics class is 45 which equal to 96th percentile. What does this mean?

a. 96% of Robert Joseph’s classmates got a score higher than 45.

b. 96% of Robert Joseph’s classmates got a score lower than 45.

c. Robert Joseph’s score is less than 45% of his classmates.

d. Roberts Joseph’s is higher than 96% of his classmates.

Example

Which one describes the percentile rank of a given score?

a. The percent of cases of a distribution below and above a given score.

b. The percent of cases of a distribution below the given score.

c. The percent of cases of a distribution above the given score.

d. The percent of cases of a distribution within the given score.

Example

Biboy obtained a score of 85 in Mathematics multiple choice tests. What does this mean?

a. He has a rating of 85

b. He answered 85 items in the test correctly

c. He answered 85% of the test item correctly

d. His performance is 15% better than the group

Example

Median is the 50th percentile as Q3 is to

a. 45th percentileb. 70th percentilec. 75th percentiled. 25th percentile

Example

Karl Vince obtained a NEAT percentile rank of 95. This means that

a. They have a zero reference pointb. They have scales of equal unitsc. They indicate an individual’s relative standing in a group

d. They indicate specific points in the normal curve

Example

Markie obtained a NEAT percentile rank of 95.

This means thata. He got a score of 95.b. He answered 95 items correctly.

c. He surpassed in performance of 95% of his fellow examinees.

d. He surpassed in performance 0f 5% of his fellow examinees.

Example

What is/are important to state when explaining percentile-ranked tests to parents?

I. What group took the testII. That the scores show how students

performed in relation to other studentsIII. That the scores show how students

performed in relation to an absolute measure

A. II only B. I & IIIC. I & II D. III only

Measures of VariationA measure of variation is a single value that is used to describe the spread of the distribution.

A measure of central tendency alone does not uniquely describe a distribution.

Mean = 15.5 s = 3.338

11 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

Section B

Section A

Mean = 15.5

s = .9258

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5 s = 4.57

Section C

A look at dispersion… Pilot Source: Training Course on Teaching Basic Statistics by Statistical Research and Training Center Philippine Statistical Association , Inc.

Two Types of Measures of Dispersion

Absolute Measures of Dispersion: Range Inter-quartile Range Variance Standard Deviation

Relative Measure of Dispersion: Coefficient of Variation

Range (R)The difference between the maximum and minimum value in a data set, i.e.

R = MAX – MINExample: Scores of 15 students in mathematics quiz. 54 58 58 60 62 65 66 71 74 75 77 78 80 82 85

R = 85 - 54 = 31

Some Properties of the Range The larger the value of the

range, the more dispersed the observations are.

It is quick and easy to understand.

A rough measure of dispersion.

Inter-Quartile Range (IQR)The difference between the third quartile and first quartile, i.e.

IQR = Q3 – Q1

Example: Scores of 15 students in mathematics quiz.

54 58 58 60 62 65 66 71 74 75 77 79 82 82 85

IQR = 78 - 61 = 17

Some Properties of IQR Reduces the influence of

extreme values.

Not as easy to calculate as the Range.

Consider only the middle 50% of the scores in the distribution

Quartile deviation (QD)is based on the range of the middle 50% of the scores, instead of the range of the entire set.

it indicates the distance we need to go above and below the median to include approximately the middle 50% of the scores.

Variance important measure of variation

shows variation about the mean

Population variance

Sample variance

Standard Deviation (SD) most important measure of variation

square root of Variancehas the same units as the original data

is the average of the degree to which a set of scores deviate from the mean value

it is the most stable measures of variability

Population SD

Sample SD

Data: 10 12 14 15 17 18 18 24 are the scores of students in mathematics quiz.

n = 8 Mean =16

309.4 7

2)1624(2)1618()1618(2)1617(2)1615(2)1614(2)1612(2)1610( 2

s

Computation of Standard Deviation

Remarks on Standard Deviation

If there is a large amount of variation, then on average, the data values will be far from the mean. Hence, the SD will be large.

If there is only a small amount of variation, then on average, the data values will be close to the mean. Hence, the SD will be small.

Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

Section B

Section A

Mean = 15.5 s = .9258

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5 s = 4.57

Section C

Comparing Standard Deviation

Example: Team A - Heights of five marathon players in inches

65”

65 “ 65 “ 65 “ 65 “ 65 “

Mean = 65 S = 0

Comparing Standard Deviation

Example: Team B - Heights of five marathon players in inches

62 “ 67 “ 66 “ 70 “ 60 “

Mean = 65” s = 4.0”

Comparing Standard Deviation

Properties of Standard Deviation It is the most widely used measure of dispersion. (Chebychev’s Inequality)

It is based on all the items and is rigidly defined.

It is used to test the reliability of measures calculated from samples.

The standard deviation is sensitive to the presence of extreme values.

It is not easy to calculate by hand (unlike the range).

Chebyshev’s Rule

It permits us to make statements about the percentage of observations that must be within a specified number of standard deviation from the mean

The proportion of any distribution that lies within k standard deviations of the mean is at least 1-(1/k2) where k is any positive number larger than 1.

This rule applies to any distribution.

Chebyshev’s Rule For any data set with mean () and standard deviation (SD), the following statements apply:

At least 75% of the observations are within 2SD of its mean.

At least 88.9% of the observations are within 3SD of its mean.

Illustration

At least 75%

At least 75% of the observations are within 2SD of its mean.

ExampleThe pre-test scores of the 125 LET reviewees last year had a mean of 70 and a standard deviation of 7 points.

Applying the Chebyshev’s Rule, we can say that:

1. At least 75% of the students had scores between 56 and 84.

2. At least 88.9% of the students had scores between 49 and 91.

Coefficient of Variation (CV)

measure of relative variation usually expressed in percent shows variation relative to mean used to compare 2 or more groups

Formula : 100%

Mean

SDCV

Comparing CVsGroup A: Average Score = 90

SD = 15 CV = 16.67%

Group B: Average Score = 92 SD = 10 CV = 10.86%

Example

Mark Erick is one-half standard deviation above the mean of his group in math and one standard deviation above English. What does this imply?

a. He excels in both English and Math.

b. He is better in Math than English.c. He does not excel in English nor in Math.

d. He is better is English than Math.

Example

Which statement about the standard deviation is CORRECT?

a. The lower the standard deviation the more spread the scores are.

b. The higher the standard deviation the less the scores spread

c. The higher the standard deviation the more the spread the scores are

d. It is a measure of central tendency

Example

Which group of scores is most varied? The group with________.

a.sd = 9b.sd = 5c.sd = 1d.sd = 7

Example

Mean is to Measure of Central Tendency as___________ is to measure of variability.

a. Quartile Deviationb. Quartilec. Correlationd. Skewness

ExampleHERE ARE TWO SETS OF SCORES: SET A : 1,2,3,4,5,6,7,8,9

SET B : 3,4,4,5,5,6,6,7,9Which statement correctly applies to the twosets of score distribution?a. The scores in Set A are more spread out than

those in set B.b. The range for Set B is 5.c. The range for Set A is 8.d. The scores in Set B are more spread out

than those in Set A.

Measure of Skewness

Describes the degree of departures of the distribution of the data from symmetry.

The degree of skewness is measured by the coefficient of skewness, denoted as SK and computed as,

SD

MedianMeanK

3S

What is Symmetry?

A distribution is said to be symmetric about the mean, if the distribution to the left of mean is the “mirror image” of the distribution to the right of the mean. Likewise, a symmetric distribution has SK=0 since its mean is equal to its median and its mode.

Measure of Skewness

SK > 0positively skewed

SK < 0negatively skewed

Areas Under the Normal Curve

Correlationrefers to the extent to which the distributions are related or associated.

the extent of correlation is indicated by the numerically by the coefficient of correlation.

the coefficient of correlation ranges from -1 to +1.

Types of Correlation1.Positive Correlationa)High scores in distribution A are associated with high scores in distribution B.

b)Low scores in distribution A are associated with low scores in distribution B.

2. Negative Correlationa)High scores in distribution A

are associated with low scores in distribution B.

b)Low scores in distribution A are associated with high scores in distribution B.

3. Zero Correlationa) No association between distribution

A and distribution B. No discernable pattern.

Positive CorrelationScience Score

Math Score

Negative CorrelationScience Score

Math Score

No CorrelationScience

Math

Example

Skewed score distribution means:a. The scores are normally distributed.

b. The mean and the median are equal.

c. Consist of academically poor students.

d. The scores are concentrated more at one end or the other end

Example

Skewed score distribution means:a. The scores are normally distributed.

b. The mean and the median are equal.

c. Consist of academically poor students.

d. The scores are concentrated more at one end or the other end

Example

What would be most likely most the distribution if a class is composed of bright students?

a. platykurtic b. skewed to the right c. skewed to the left d. very normal

Example

All the students who took the examination, got

scores above the mean. What is the graphical

representation of the score distribution?

a. normal curveb. mesokurticc. positively skewedd. negatively skewed  

A class is composed of academically poor students. The distribution most likely to be______________.

a. skewed to the rightb. a bell curvec. leptokurticd. skewed to the left

Z-SCORE

In statistics, a standard score (also called z-score) is a dimensionless quantity derived by subtracting the population mean from an individual (raw) score and then dividing the difference by the population standard deviation.

The Z-score reveals how many units of the standard deviation a case is above or below the mean. The z-score allows us to compare the results of different normal distributions, something done frequently in research.

whereX is a raw score to be standardizedσ is the standard deviation of the

populationµ is the mean of the population

The quantity z represents the distance between the raw score and the population mean in units of the standard deviation. z is negative when the raw score is below the mean, positive when above.

The Standard score is :

A key point is that calculating z requires the population mean and the population standard deviation, not the sample mean or sample deviation. It requires knowing the population parameters, not the statistics of a sample drawn from the population of interest.N) T-SCORE

it is equivalent to ten times the Z-score plus fifty

T=10Z + 50

EXAMPLE: Based on the table shown, who performed better, JR or JM? Assume a normal distribution.

Student Raw Score Mean Standard DeviationJR 75 65 4JM 58 52 2

For JR

For JM

JM performed better than JR due to a greater value of z.

From the previous example, the T-score of JR is

T JR = 10(2.5) + 50 = 75

While the T-score of JM is

T JM = 10(3) + 50 = 80Therefore, JM performed better than JR due to higher T-score

O) STANINEStanine (Standard NINE)

Is a method of scaling test scores on a nine-point standard scale in a normal distribution.

Percentage Distribution

4% 7% 12% 17% 20% 17% 12% 7% 4%

Cumulative Percentage Distribution

4% 11% 23% 40% 60% 77% 89% 96% 100%

STANINE 1 2 3 4 5 6 7 8 9

ExampleStudy this group of test which was administered to a class to whom Jar-R belongs, then answer the question:

Subject Mean SD Jay-R’s Score

Math 56 10 43Physics 55 9.5 51English 80 11.25

88PE 75 9.75 82

In which subject (s) did Jay-R perform most poorly in relation to the group’s mean performance?

A. EnglishB. PhysicsC. PE D. Math

Based on the data given , what type of learner is Jay-R?

A. LogicalB. SpatialC. LinguisticD. Bodily-Kinesthetic

Based on the data given , in which subject (s) were scores most widespread?

A. MathB. PhysicsC. PED. English

ReferencesPilot Training Course on Teaching Basic

Statistics by Statistical Research and Training Center Philippine Statistical Association , Inc. (Power point presentation on the different concepts of Statistics)

Elementary Statistics by Yonardo A. Gabuyo et. al. Rex Book Store

Assessment of Learning I and II by Dr. Rosita De Guzman-Santos, LORIMAR Publishing, 2007 Ed.

Measurement and Evaluation Concepts and Principles by Abubakar S. Asaad and Wilham M. Hailaya, Rex Book Store

LET Reviewer by Yonardo A. Gabuyo, MET Review Center