Chapter IChapter IMeasures of Central TendencyMeasures of Central Tendency

& Variability& Variability

Curriculum Objective:•The students will determine the measures of central tendency and variability•Apply these tendencies to solving problems• Analyze these measure in the case

Descriptive Inferential

Statistics

What is Statistics?

• Descriptive Statistics

Describe the characteristic of the data

such as ; mean, median, std dev, variansi etc

• Inferential Statistics

Make an inferences about the population, characteristics from information contained in a sample drawn from this population

Such as : prediction, estimation, take the decision

Mean

Median

Mode

Measures ofCentral Tendency

Variance

Standard Deviation

Range

Deviation

Mean Deviation

Sum of SquaredDeviation

Measure ofVariability

FrequencyDistribution tables

FrequencyDistribution Polygon

Histogram

Bar Graph

GraphicDisplays

DescriptiveStatistics

z-scores

Single Samplet statistic

Independentt statistic

Dependentt statistic

Estimation

PearsonCorrelation

Phi-coefficient

Linear Regression

Correlation &Regression

Test forGoodness of Fit

Test forIndependence

Chi-Square

InferentialStatistics

1. PopulationIs the set of all measurements of interest the

investigator parameter

2.SampleIs a subset of measurements selected from

the population of interest statistic

Data Scale

Qualitative Data

a.  Nominal

Example: gender, date birth same level

b. Ordinal

Example : taste, grade score(difference level) Quantitative Data a.   Interval Data have a range Example : Hot enough: 50 – 80 derajat C, Hot 80 – 110 C, Very Hot: 110 – 140 C b. Ratio Data Can be applied with mathematic operations

Example : height, weight

An Naas

Dispersion tendency

MISSING

AIM

Central tendencyCentral tendencyQOLBQOLB

Dispersion tendency

MISSING

Dispersion tendency

MISSING

Statistic Ilustration

• Imagine you were a statistician, confronted with a set of numbers like 1,2,7,9,11

• Consider a notion of “location” or “central tendency – the “best measure” is a single number that, in some sense, is “as close as possible to all the numbers.”

• What is the “best measure of central tendency”?

Measure of central tendency

• Central tendencyCentral tendency– A statistical measure that identifies A statistical measure that identifies

a single score as representative for a single score as representative for an entire distribution. an entire distribution.

– The goal of central tendency is to The goal of central tendency is to find the single score that is most find the single score that is most typical or most representative of typical or most representative of the entire group.the entire group.

Measure of central tendencyMeasure of central tendency1.1. MeanMeanPopulation mean vs. sample mean

– Example

N=4: 3,7,4,6

N

X

54

20

n

Xx

n

Xx

Computing the Mean from a Frequency Distribution

X f

30 2

29 3

28 5

27 3

26 2

1

1

K

i iiN

ii

X fX

f

Estimating the Mean from a Grouped Frequency Distribution

Example

Interval f MdPt Sum

81-90 7 85.5 598.5

71-80 11 75.5 830.5

61-70 4 65.5 262.0

51-60 3 55.5 166.5

25 1857.5

2. Median– The score that divides a distribution exactly in

half. – Exactly 50 percent of the individuals in a

distribution have scores at or below the median.

– The median is often used as a measure of central tendency when the number of scores is relatively small, when the data have been obtained by rank-order measurement, or when a mean score is not appropriate.

– Therefore, it is not sensitive to outliers

Calculating the MedianCalculating the Median

• Order the numbers from highest to lowest• If the number of numbers is odd, choose the

middle value• If the number of numbers is even, choose the

average of the two middle values.– odd: 3, 5, 8, 10, 11 median=8– even: 3, 3, 4, 5, 7, 8 median=(4+5)/2=4.5

Note :Note :The mean is “sensitive to outliers,” while the median is not.The mean is “sensitive to outliers,” while the median is not.

Sensitivity to OutliersEx: Incomes in Weissberg, Nova Scotia (population =5)

Person Income (CAD)

Sam 5,467,220

Harvey 24,780

Fred 24,100

Jill 19,500

Adrienne 19,400

Mean 1,111,000

In the above example, the mean is $1,111,000, the median is In the above example, the mean is $1,111,000, the median is 24,100.24,100.Which measure is better?Which measure is better?

Mean : Sensitivity to OutliersMean : Sensitivity to Outliers

Incomes in Weissberg, Nova Scotia (population =5)

In the above example, the mean is $1,111,000, the median is 24,100. Which measure is better?

Person Income (CAD)

Sam 5,467,220

Harvey 24,780

Fred 24,100

Jill 19,500

Adrienne 19,400

Mean 1,111,000

04/19/23

FREQUENCIES DISTRIBUTION : It used to organized sistematically data in many group without reduce the data information

If there are a lot of data then its will be divide on many of class but if the there are little data then we need’nt to devide it

04/19/23

Steps to make freq. distr

• Decide the amount of the class (∑K) that will taken from N data

• Decide the range

• Decide Class Interval

∑K = 1 + 3,3 Log N

Range (R) = The biggest obs-the smallest obs

Ci = R / ∑K

Example

∑K = 1 + 3,3 Log 80 =7.28~7

R=99-35=64

Ci=64/7=9.14 ~10

Finding Mean for grouped dataFinding Mean for grouped data

with

xk=midle value every classnilai tengah tiap kelas

fk =class frequencies

withTb : lb med interval class,i: class intervalN : amount of the observationfseb : cum freq before med class

med

seb2

1

medf

fNi

Tb

Finding Med For grouped dataFinding Med For grouped data

MODUS

Is value or phenomenon that the most often appear, if the data is grouped than we have :