STATISTICAL TOOLS NEEDED

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• STATISTICAL TOOLSNEEDED
• IN ANALYZING TEST RESULTSProf. Yonardo
  Agustin Gabuyo

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Statistics is a branch of science which deals
with the collection, presentation, analysis and
interpretation of quantitative data.
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Branches of Statistics

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

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• 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.

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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.

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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.

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• 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

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• Example
• This is a type of statistics that give/s
  information about the sample being studied.
• a. Inferential and co-relational
• b. Inferential
• c. Descriptive
• d. Co relational

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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
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Types of Variables
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Qualitative variables? variables that can be
express in terms of properties, characteristics,
or classification(non-numerical values).
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Quantitative Variables ? variables that can
be express in terms of numerical
values.a)Discrete- variables that can be express
in terms of whole number.b)Continuous-
variables that can be express in terms whole
number, fraction or decimal number.
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Levels of Measurement

• Nominal
• ? Numbers or symbols used to classify
• Ordinal scale
• ? Accounts for order no indication of distance
  between positions
• Interval scale
• ? Equal intervals no absolute zero
• Ratio scale
• ? Has absolute zero

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Methods of Collecting Data

• Objective Method

• Subjective Method

• Use of Existing Records

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Methods of Presenting Data

• ? Textual
• ? Tabular
• ? Graphical

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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, Maximum
• Central Tendency
• ?Percentiles, Deciles, Quartiles

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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.

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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 simple
• most representative value of the data

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Mean

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

Population Mean
Sample Mean
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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.

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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
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Mean 6
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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

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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
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Median 5
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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
0 1 2 3 4 5 6 7 8 9 10 11
12 13 14
No Mode
Mode 9
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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

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Mean, Median Mode

• Use the mean when
• ? sampling stability is desired
• ? other measures are to be computed

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Mean, Median Mode

• Use the median when
• ? the exact midpoint of the distribution is
  desired
• ? there are extreme observations

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Mean, Median Mode

• Use the mode when
• ? when the "typical" value is desired
• ? when the dataset is measured on a nominal scale

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• Example
• Which measure(s) of central tendency is(are)
  most appropriate when the score distribution is
  skewed?
• A. Mode
• B. Mean and mode
• C. Median
• D. Mean

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• Example
• In one hundred-item test, what does Jay-Rs 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 mean
• D. He got 70 items correct

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• Example
• Which of the following measures is more affected
  by an extreme score?
• A. Semi- inter quartile range
• B. Median
• C. Mode
• D. Mean

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• 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 N
• d. The mean divided by N
•  

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• 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. Mode
• B. Mean
• C. Median
• D. Mean and Median

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• 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?
• Mode only
• Mode and median
• c. Median only
• d. Mean and mode

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• Example
• What is the mean of this score distribution 40,
  42, 45, 48, 50, 52, 54, 68?
• a. 51.88
• b. 50.88
• c. 49.88
• d. 68

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• Example
• Which is the correct about MEDIAN?
• a. It is measure of variability
• b. It is the most stable measure of central
  tendency
• c. It is the 50th percentile
• d. It is significantly affected by extreme values

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• Example
• Which measure(s) of central tendency can be
  determined by mere inspection?
• a. Median
• b. Mode
• c. Mean
• d. Mode and Median
•  

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• 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-modal
• C. Skewed to the right D. No discernible pattern

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• 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.

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• Example
• STUDY 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

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• In which scores interval is the median?
• a. In the interval 80 to 89
• b. In between the intervals of 60-69 and 70-79
• c. In the interval 70-79
• d. In the interval 60-69

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• How many percent of the students got a score
  below 70?
• a. 2
• b. 8
• c. 10
• d. 39

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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 students raw score.

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• ?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).

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EXAMPLE

• Suppose 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 JMs score, while 3 had
  scores higher than JMs.

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Deciles

• ?Divides 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.

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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.

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• Example
• Robert Josephs raw score in the mathematics
  class is 45 which equal to 96th percentile. What
  does this mean?
• a. 96 of Robert Josephs classmates got a score
  higher than 45.
• b. 96 of Robert Josephs classmates got a score
  lower than 45.
• c. Robert Josephs score is less than 45 of his
  classmates.
• d. Roberts Josephs is higher than 96 of his
  classmates.

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• 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.

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• 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

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• Example
• Median is the 50th percentile as Q3 is to
• a. 45th percentile
• b. 70th percentile
• c. 75th percentile
• d. 25th percentile

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• Example
• Karl Vince obtained a NEAT percentile rank of
  95. This means that
• a. They have a zero reference point
• b. They have scales of equal units
• c. They indicate an individuals relative
  standing in a group
• d. They indicate specific points in the normal
  curve

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• Example
• Markie obtained a NEAT percentile rank of 95.
• This means that
• a. 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.

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• Example
• What is/are important to state when explaining
  percentile-ranked tests to parents?
• I. What group took the test
• II. That the scores show how students performed
  in relation to other students
• III. That the scores show how students performed
  in relation to an absolute measure
• A. II only B. I III C. I II D. III only

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Measures of Variation

• A 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.

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A look at dispersion Pilot Source Training
Course on Teaching Basic Statistics by
Statistical Research and Training Center
Philippine Statistical Association , Inc.
Section A
Mean 15.5 s 3.338
11 12 13 14 15 16 17 18
19 20 21
Section B
Mean 15.5 s .9258
11 12 13 14 15 16 17 18
19 20 21
Section C
Mean 15.5 s 4.57
11 12 13 14 15 16 17 18
19 20 21
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Two Types of Measures of Dispersion

• Absolute Measures of Dispersion
• ? Range
• ? Inter-quartile Range
• ? Variance
• ? Standard Deviation

Relative Measure of Dispersion ? Coefficient
of Variation
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Range (R)

• The difference between the maximum and minimum
  value in a data set, i.e.
• R MAX MIN

Example 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
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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.

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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
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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

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• 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.

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Variance

• ? important measure of variation
• ? shows variation about the mean
• Population variance
• Sample variance

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Standard Deviation (SD)

• ? most important measure of variation
• ? square root of Variance
• has 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

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Population SD Sample SD
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Computation of Standard Deviation
Data 10 12 14 15 17 18 18
24 are the scores of students in mathematics
quiz.
n 8 Mean 16
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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.
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Comparing Standard Deviation
Section A
Mean 15.5 s 3.338
11 12 13 14 15 16 17 18
19 20 21
Section B
Mean 15.5 s .9258
11 12 13 14 15 16 17 18
19 20 21
Section C
Mean 15.5 s 4.57
11 12 13 14 15 16 17 18
19 20 21
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Comparing Standard Deviation
Example Team A - Heights of five marathon
players in inches

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Mean 65 S 0
65
65
65
65
65
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Comparing Standard Deviation
Example Team B - Heights of five marathon
players in inches

Mean 65 s 4.0
62
67
66
70
60
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Properties of Standard Deviation

• ? It is the most widely used measure of
  dispersion. (Chebychevs 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).

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Chebyshevs 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.

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Chebyshevs 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.

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

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

Applying the Chebyshevs 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.
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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

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Comparing CVs

• Group A Average Score 90
• SD 15
• CV 16.67
• Group B Average Score 92
• SD 10
• CV 10.86

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• 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.

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• 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

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• Example
• Which group of scores is most varied? The group
  with________.
• sd 9
• sd 5
• sd 1
• sd 7

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• Example
• Mean is to Measure of Central Tendency
  as___________ is to measure of variability.
• a. Quartile Deviation
• b. Quartile
• c. Correlation
• d. Skewness

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• Example
• HERE ARE TWO SETS OF SCORES
• SET A 1,2,3,4,5,6,7,8,9
• SET B 3,4,4,5,5,6,6,7,9
• Which statement correctly applies to the two
• sets 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.

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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,

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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 SK0 since its mean is equal to
  its median and its mode.

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Measure of Skewness

• SK gt 0
• positively skewed

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• SK lt 0
• negatively skewed

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Areas Under the Normal Curve
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• Correlation
• ?refers 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.

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• Types of Correlation
• Positive Correlation
• High scores in distribution A are associated with
  high scores in distribution B.
• Low scores in distribution A are associated with
  low scores in distribution B.

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• 2. Negative Correlation
• High scores in distribution A are associated with
  low scores in distribution B.
• Low scores in distribution A are associated with
  high scores in distribution B.
• 3. Zero Correlation
• a) No association between distribution A and
  distribution B. No discernable pattern.

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Positive Correlation
Science Score
Math Score
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Negative Correlation
Science Score
Math Score
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No Correlation
Science
Math
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• 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

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• 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

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• 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

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• Example
• All the students who took the examination, got
• scores above the mean. What is the graphical
• representation of the score distribution?
• a. normal curve
• b. mesokurtic
• c. positively skewed
• d. negatively skewed
•  

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• A class is composed of academically poor
  students. The distribution most likely to
  be______________.
• a. skewed to the right
• b. a bell curve
• c. leptokurtic
• d. skewed to the left

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• 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.

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The Standard score is
where X is a raw score to be standardized s is
the standard deviation of the population µ 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.
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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 T10Z 50
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EXAMPLE Based on the table shown, who performed
better, JR or JM? Assume a normal
distribution. Student Raw Score Mean
Standard Deviation JR 75
65 4 JM 58 52 2 For JR
For JM JM performed better than JR due
to a greater value of z.
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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
80 Therefore, JM performed better than JR due to
higher T-score
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O) STANINE Stanine (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
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• Example
• Study this group of test which was administered
  to a class to whom Jar-R belongs, then answer the
  question
• Subject Mean SD Jay-Rs Score
• Math 56 10 43
• Physics 55 9.5 51
• English 80 11.25 88
• PE 75 9.75 82

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• In which subject (s) did Jay-R perform most
  poorly in relation to the groups mean
  performance?
• A. English
• B. Physics
• C. PE
• D. Math

108

• Based on the data given , what type of learner is
  Jay-R?
• A. Logical
• B. Spatial
• C. Linguistic
• D. Bodily-Kinesthetic

109

• Based on the data given , in which subject (s)
  were scores most widespread?
• A. Math
• B. Physics
• C. PE
• D. English

110

• References
• Pilot 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