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Centre for Computer Technology

ICT114Mathematics for

Computing

Week2

Statistics and FrequencyDistribution


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March 20, 2012 Copyright Box Hill Institute

Set : Introduction

A set is a well-defined list, collection or class of objects.

The objects could be anything : numbers, names,

people, cities. These objects are called the elements ormembers of the set.

Example 1: The numbers 1,3,5,7,9,11,13,Example 2: The solutions of the equation x2 4x+3=0Example 3 : The rivers in Australia


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Set Notation

Sets are usually denoted by capital letters

A, B, P, X, ..

The elements are usually represented bylowercase letters a, b, p, x, ..

There are two forms for presentation of a set :Tabular form , A = {1,3,5,7,9,11,}Set builder form, A = {x | x is odd}


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Subsets

If every element in a set A is also a member of aset B, then A is called a subsetof B

In other words,

if x A x B for all x,

then A is a subsetof B

It is written as AB or BA

A is called a proper subsetof B, if A B and Ais not equal to B.


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Venn Diagram to represent sets

U is the universal set.

A and B are disjoint sets

R is a subset of S

UU

A

B

S

R


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Set Operations

Let A and B represent two sets. We havethe definitions in a compact manner

1. A U B ={ x | x

A or x

B or x

both}2. A B ={ x | x A and x B }3. A B ={ x | x A and x B }

4. A

/

={ x | x

A }5. A B={ x | x A or x B but x both}6. #A = Number of elements in set A


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Centre for Computer Technology

Statistics and FrequencyDistribution


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Introduction

Statistics is the medium to describe thecenter spread and shape of a data set.

Two components Gathering of information or scientific data

Inferential statistics/Statistical methods

Statistical Methods are employed to makejudgements in the face of uncertainty andvariation.


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Measures of Central Tendency Measures of Central Tendency are single

values that act as a representative of data

Three main measures

Mean

Mode

Median


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Mean

For a given set of n numbers x1,x2,x3,.....xn.

The mean denoted by

x1+x2+x3+.....+xn

= ------------------------

n


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Example : Consider the following set of

numbersS = {1, 2, 3, 4, 5, 6, 7, 8, 9}

The mean of the set S is

1+2+3+4+5+6+7+8+9

= ------------------------------- = 59


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Median

For a given set of n numbers x1,x2,x3,.....xn

Median is a value where half the values

are of x1,x2,x3,.....xn are larger than themedian and the other half are smaller thanthe median.

In other words, Median is the middlemostnumber


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Median

Example : Consider the following set ofnumbers

S = {1, 6, 3, 8, 2, 4, 9}

To find the median, we need to order thelist

S = {1, 2, 3, 4, 6, 8, 9}

The middlemost number is 4 which is themedian of the set.


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March 20, 2012Copyright Box Hill Institute

What happens when we have to find the

median of a set with an even number ofelements

For example:

Find the median of

S = {1, 6, 3, 8, 2, 12, 4, 9}


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Some More Concepts

For a set of n ordered data points

If n is odd, the median is found in the location(n+1)/2 of the set

If n is even, the median is the average of thetwo middle terms.

The two terms are found in the location

n/2, n/2+1


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Mode

Mode of a data set is the value that occursmost often

If there are two, three or multiple valuesthe data is bimodal, trimodal or multimodal

Example:

R = {2, 8, 1, 9, 5, 2, 7, 2, 7, 9, 4, 7, 1, 5, 2}

The number that appears most is 2, whichis the mode of R.


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Measures of Dispersion

Consider two sets

S={5, 5, 5, 5, 5, 5}

R={0, 0, 0, 10, 10, 10}for both the above sets, mean = 5

But the above sets are two different datasets.

Is it a good practice to use mean, medianor mode to describe them?


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We use another descriptive statistic toevaluate the data called Measure of

Dispersion. It is a measure of scatter or

dispersion.

It is a measure of scatter about the

mean.


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What happens to the values of dispersion

If they are concentrated near the mean ?

If they are distributed far from the mean?


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If the values are concentrated near the meanof the data set, the measure is small.

If they are distributed far from the mean of thedata set, the measure will be large.

There are two main measures of dispersion

Variance

Standard Deviation


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Variance and Standard Deviation

For a given set of n numbersx1,x2,x3,.....xn, the Variance, denoted by 2

is given by

(x1- )2 + (x2- )2 + .....+ (xn- )2

2 = -------------------------------------------n


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Variance (method 2)

Variance (method 2)

= Mean of squares minusSquare ofMean

= ( x2 / n) - ( x / n)2

=

x = x1 + x2 + x3........+ xn


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The Variance is a non negative number

The positive square root of the varianceis standard deviation.

The simplest spread of variability isSample Range.Xmax - Xmin


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Example: Find the variance and standarddeviation for the following set of test scores:

T = {75, 80, 82, 87, 96}

The mean of the set T is

75+80+82+87+96 = ------------------------------- = 845


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March 20, 2012

Copyright Box Hill Institute

Using the mean we get the variance as

(75-84)2 + (80-84)2 + (82-84)2 + (87-84)2 + (96-84)2

2 = ----------------------------------------------------

5

= 50.8

Standard Deviation = 2 = 7.1274


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March 20, 2012

Copyright Box Hill Institute

Sample Space

Set of all possible outcomes of a

statistical experiment is called a sample

space or sample Each outcome is called an element or a

member or sample point

A group of samples is called population


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Sample Statistics

Any quantity obtained from a sample forthe purpose of estimating a populationparameter is called a sample statistic

A sample along with inferential statisticsallow us to draw conclusions about

population, with inferential statisticsmaking clear use of elements ofProbability.


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Sample Mean

For a given sample of n numbersx1,x2,x3,.....xn.

The sample mean denoted by X

x1+x

2+x

3+.....+x

n

X = ------------------------

n


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Weighted Mean

For a given set of data, X= { x1, x2, ..., xn}

and corresponding non-negative weights,

W= { w1, w2, ..., wn}the weighted mean/average, is given by

w1x1+w2x2+w3x3+.....+wnxn

X = ---------------------------------------w1+w2+w3++wn


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Sample Variance

For a given sample of n numbersx1,x2,x3,.....xn, the Variance, denoted by S

2

is given by

(x1- X)2 + (x2- X)

2 + .....+ (xn- X)2

S2 = -------------------------------------------(n-1)


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Frequency Distributions

For large samples (or populations) it isdifficult to observe various characteristics

or to compute statistics Therefore it is useful to organize or group

the raw data

The data is arranged in intervals of equalwidth.


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Frequency Distributions

The intervals are called classes orcategories.

The number of individuals or elements ineach class is determined, called classfrequency.

The resulting arrangement is calledfrequency distribution or frequency table.


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Frequency Distribution

Example : Height ofstudents in XYZ

university (frequencytable)

Height

(cm)

Number of

Students

155-159

160-164

165-169

170-174

175-179

5

18

42

27

8

Total 100


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Frequency Distribution

In the previous example

The first category 155-159 is called classinterval

The corresponding class frequency is 5.

The mid point of the class interval is calledthe class mark.


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Frequency Histogram

Height

(cm)

Number of

Students

155-159

160-164

165-169

170-174

175-179

5

18

42

27

8

Total 100 0

5

10

15

20

25

30

35

40

45

155-159 160-164 165-169 170-174 175-179

Height (cm)


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Frequency Polygon

Height

(cm)

Number of

Students

155-159

160-164

165-169

170-174

175-179

5

18

42

27

8

Total 1000

5

10

15

20

25

30

35

40

45

157 161 167 172 177

Height (cm)


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Frequency Graphs

In a histogram, the sum of therectangular areas is 100.

A frequency polygon is a graphconnecting the midpoints of the topsof the histogram.

In a bar graph, the sum of theordinates is 1.


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Relative Frequency

Height

(cm)

Number of

Students

155-159

160-164

165-169

170-174

175-179

05%

18 %

42 %

27 %

08 %

Total 100%

In relative frequency,the class frequency isreplaced by

percentage ratherthan the number.

In the histogram thevertical axis will bereplaced with relativefrequency instead offrequency.


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In the previousexample, whathappens if we have a

student with a heightof 159.7 cm.

Height

(cm)

Number of

Students

155-159

160-164

165-169

170-174

175-179

Total


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Continuous Frequency Distribution

The class intervalsare chosen such thatthey are continuous

as shown

Height

(cm)

Number of

Students

154.5-159.4

159.5-164.4

164.5-169.4

169.5-174.4

174.5-179.4

Total


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Mean and Variance from

Frequency TableInterval mid point (x) frequency (f) f.X f.X2

a0- a1 x1 f1 f1.x1 f1.x1.x1

a1- a2 x2 f2 f2.x2 f2.x2.x2

an-1an xn fn fn.xn fn.xn.xn

All Total f Total f.x Total f.x.x

Mean = total (f.x) / total f Variance = [total (f.x.x)/total f] (mean)2


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Example : Mean and Variance from

Frequency TableClass interval Frequency, f

1.5 1.9

2.0 2.4

2.5 2.9

3.0 3.4

3.5 3.94.0 4.4

4.5 4.9

2

1

4

15

10

5

3


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Class interval Class

midpoint, x

Frequency, f f.x f.x2

1.5 1.9

2.0 2.4

2.5 2.9

3.0 3.4

3.5 3.9

4.0 4.4

4.5 4.9

1.7

2.2

2.7

3.2

3.7

4.2

4.7

2

1

4

15

10

5

3

3.4

2.2

10.8

48

37

21

14.1

5.78

4.84

29.16

153.6

136.9

88.2

66.2740 136.5 484.75


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Mean = total (f.x) / total f

= 136.5 / 40

= 3.4125

Variance = [total (f.x.x)/total f] (mean)2

= 484.75 / 40 (3.4125)2

= 12.1188 11.6452= 0.4736


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Summary

There are three main measures of centraltendency : Mean, Mode and Median.

There are two main measures ofdispersion : Variance and StandardDeviation.

The organization or grouping of raw datain a table is called Frequency distribution.


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References

M R Spiegel : Theory and Problems ofStatistics, Schaum's Outline Series,McGraw Hill.

http://mathworld.wolfram.com

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