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Statistics plays a very
important role in measurement andevaluation. Statistical techniquesallow us to describe the
performance of our students andmake proper scientific inferencesabout their performance. This
chapter introduces the students tosome very basic concepts in
Statistics.
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Statistics is a scientific body of
knowledge that deals with:
Collection of dataOrganization or presentation ofdata
Analysis and interpretation ofdata
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Introduction
toStatisticalConcepts
The Idea ofRandomnes
Measures ofCentral
Tendency
TheStandard
Deviation
FrequencyDistribution
Table
Measures ofLocation and
Fractiles
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Populationidentifies the totality of(P) objects under investigation.
refers to a large collectionof objects, places or things.
Samplesmall portion or part of a(S) population.
- a representative of a
population in a research study.
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Population
Sample
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Let us consider a group of thirty (30)students whose examination scores are as
follows:P={15,20,25,25,30,35,26,14,17,18,12,15,22,27,33,3
2,21,18,18,13,14,21,17,18,19,23,22,21,26,31}
Now, the teacher randomly selects fifteen(15)examination scores for further analysis andobtains:S= {20,15,14,12,33,18,21,17,22,21,26,30,32,13,15}
The idea of randomnessunpredictability.
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While the mean is a good singledescription of the location of the central
tendency of the data, it is often not enough tofully describe the data sets. A measure ofdispersion or scatter is also needed. The
standard deviation is a measure of theaverage deviation or departure of theindividual scores from the mean and is given
by:
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Mean (or average) is equal to the sum of
all values in the data set divided by the numberof values in the data set.
Mean of PopulationP= (15++31)/ 30 = 648/30 = 21.60
Mean of SampleS= (20+ +15)/15 = 319/15 = 21.27
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For grouped data:
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Median- is the set of numbersarranged in order of magnitude is either themiddle value or the arithmetic mean of the2 middle values.Where:
LB- lower bound of the middle classn- the number of the observations
F- cumulative frequency of the class previous
to the middle class
f- is the frequency of the middle class- the size of the modal class interval
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Mode - is the value which occurs withthe greatest frequency.
for grouped data:Where:
LB- Lower bound of the middle class
d1- the difference between the modalfrequency and the frequency of the next
lower class
d2- is the difference between the modal
frequency and the frequency of the nexthighest class.
- the size of the modal class interval
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Steps in Constructing Frequency Distribution TableStep 1: Determine the range
R = Highest Value Lowest Value
Step 2: Determine the tentative number of classes (k)k = 1+3.322 log N - always round off
Step 3: Find the class width by dividing the range by the
number of classes.Class width = range/ number of classes or c = R/k
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77,77,78,80,80,80,80,81,82,82,82,82,83,83,83,83,83,84,84,84,84,84,84,84,84,87,87,87,87,87,88,
88,88,88,88,88,89,89,89,89,91,91,91,91,93,94,94,94,95,97
R = Highest value lowest value
= 97-77R = 20
k = 1 + 3.322logN= 1 + 3.322 log50
= 1 + 5.6
k = 6.64 or 7
Class width = R/k
= 20/7C= 2.86 or 3
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(Final Grade) (f) (CB) (x)
77
79 3 76.5
79.5 78
80 82 9 79.5 82.5 81
83 85 13 82.5 85.5 84
86 88 11 85.5 88.5 87
89 91 8 88.5 91.5 90
92
94 4 91.5
94.5 93
95 97 2 94.5 97.5 96
f = 50
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(Final
Grade)(f) (CB) (x) (
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MEAN
=
M = 85.92
MEDIAN
=
Md = 85.5MODE
=
MO = 84.5
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Measures of Location or
Fractiles- values below which a specified fractionor percentage of the observations in a given setmust fall.
Percentiles (100)Deciles (10)Quartiles (4)
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Percentilesare values that divide a set ofobservations in an array into 100 equal parts.
For ungrouped:
Decilesare values that divide a set of
observations in an array into 10 equal parts.For ungrouped:
Quartilesare values that divide a set ofobservations in an array into 4 equal parts.
For ungrouped:
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Example: the ff. were the scores of 10
students. 2, 8, 6, 9, 7, 5, 8, 10, 10, 1. Findthe 64th percentile
= = 7.04 or 8
1, 2, 5, 6, 7, 8, 8, 9, 10, 10
64% of the scores are below 9
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Percentile from a Frequency Distribution
Where:
the Pith class is the class where the
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Age Frequency
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