central tendency and variance

7/22/2019 Central Tendency and Variance

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1Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Measures of Central Tendency andVariation




Measures of Center or

Average

a value at the center or middle of a data set

Significance: Single value to represententire dataset eg. average pay package of

a MBA institute




Mean or Arithmetic Mean

MedianMode

Mid Range

Different Measures of Center orAverage




Mean

the number obtained by adding thevalues and dividing the total by thenumber of values

Definitions




Notation

denotes the addition of a set of values

x is the variable usually used to represent the individual

data values

n represents the number of data values




Notation

is pronounced ‘x-bar’ and denotes the mean of a setof values

x =n

x ix


http://slidepdf.com/reader/full/central-tendency-and-variance 7/327Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Data values:6.70 3.46 3.60 6.44

6.70+3.46+3.60 + 6.44

4Mean is 5.05



Definitions

Medianthe middle value when the originaldata values are arranged in order of

increasing (or decreasing) magnitude



Definitions



often denoted by x (pronounced ‘x-tilde’) ~



Definitions



often denoted by x (pronounced ‘x-tilde’)

is not affected by an extreme value

~



6.72 3.46 3.60 6.44

3.46 3.60 6.44 6.72

no exact middle -- shared by two numbers

3.60 + 6.44

2

(even number of values)

MEDIAN is 5.02



6.72 3.46 3.60 6.44 26.70

3.46 3.60 6.44 6.72 26.70

(in order - odd number of values)

exact middle MEDIAN is 6.44

6.72 3.46 3.60 6.44

3.46 3.60 6.44 6.72

no exact middle -- shared by two numbers

3.60 + 6.44

2

(even number of values)

MEDIAN is 5.02



Definitions

Modethe score that occurs most frequently

Bimodal

Multimodal

No Mode

denoted by M the only measure of central tendency that can be

used with nominal data



a. 5 5 5 3 1 5 1 4 3 5

b. 1 2 2 2 3 4 5 6 6 6 7 9

c. 1 2 3 6 7 8 9 10

Examples

Mode is 5

Bimodal - 2 and 6

No Mode



Midrange

the value midway between thehighest and lowest values in theoriginal data set

Definitions



Midrange

the value midway between thehighest and lowest values in theoriginal data set

Definitions

Midrange = highest score + lowest score

2



Data values:3.46 3.60 6.44 6.70

3.46 + 6.70

2 Mid Range is 5.08



use class midpoint of classes for variable x

Mean from a Frequency Table orGrouped Data

x i = i th class midpoint

f i = i th class frequency

f i = n

x = f i

(f i • x i )



Daily Petrol Consumption in Litrefor 52 Office Goers

2 2 5 1 2 6 3 3 4 2 4 0 5 7 7 5 6 6 8 10 7 2 2 10 5 8 2 5 4 2

6 2 6 1 7 2 7 2 3 8

1 5 2 5 2 14 2 2 6 3

1 7



Frequency Table of Marks

0 - 2 20

3 - 5 14

6 - 8 15

9 - 11 2

12 - 14 1

Rating Frequency



Lower Class Limitsare the smallest numbers that belong to different

classes

0 - 2 203 - 5 14

6 - 8 15

9 - 11 2

12 - 14 1

Rating Frequency




Lower Class Limits

Lower ClassLimits

0 - 2 203 - 5 14

6 - 8 15

9 - 11 2

12 - 14 1

Rating Frequency

are the smallest numbers that can actually belong to

different classes




Upper Class Limits

Upper ClassLimits

0 - 2 20

3 - 5 14

6 - 8 15

9 - 11 2

12 - 14 1

Rating Frequency

are the largest numbers that can actually belong to

different classes




are the numbers used to separate classes, butwithout the gaps created by class limits

Class Boundaries




number separating classes

Class Boundaries

0 - 2 20

3 - 5 14

6 - 8 15

9 - 11 2

12 - 14 1

Rating Frequency- 0.5

2.5

5.5

8.5

11.5

14.5




Class Boundaries

ClassBoundaries

0 - 2 20

3 - 5 14

6 - 8 15

9 - 11 2

12 - 14 1

Rating Frequency- 0.5

2.5

5.5

8.5

11.5

14.5

number separating classes




midpoints of the classes

Class Midpoints

Class

Midpoints

0 - 1 2 20

3 - 4 5 14

6 - 7 8 15

9 - 10 11 2

12 - 13 14 1

Rating Frequency




Class Width

Class Width

3 0 - 2 203 3 - 5 14

3 6 - 8 15

3 9 - 11 2

3 12 - 14 1

Rating Frequency

is the difference between two consecutive lower

class limits or two consecutive class boundaries




1. Be sure that the classes are mutually exclusive.

2. Include all classes, even if the frequency is zero.

3. Try to use the same width for all classes.

4. Select convenient numbers for class limits.

5. Use between 5 and 20 classes.

6. The sum of the class frequencies must equal thenumber of original data values.

Guidelines For Frequency Tables

C t ti A F T bl




3. Select for the first lower limit either the lowest score or aconvenient value slightly less than the lowest score.

4. Add the class width to the starting point to get the second lower

class limit, add the width to the second lower limit to get the

third, and so on.

5. List the lower class limits in a vertical column and enter the

upper class limits.

6. Represent each score by a tally mark in the appropriate class.

Total tally marks to find the total frequency for each class.

Constructing A Frequency Table1. Decide on the number of classes .

2. Determine the class width by dividing the range by the numberof classes (range = highest score - lowest score) and round up.

class width round up ofrange

number of classes




Symmetric

Data is symmetric if the left half of itshistogram is roughly a mirror of itsright half.

Skewed

Data is skewed if it is not symmetricand if it extends more to one side thanthe other.

Mean vs Median vs Mode



Skewness

Mode = Mean = Median

SKEWED LEFT

(negatively)

SYMMETRIC

Mean Mode

Median

SKEWED RIGHT

(positively)

MeanMode

Median

Figure2-13 (b)

Figure2-13 (a)

Figure2-13 (c)

central tendency and variance

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