Educational Statistics

Educational StatisticsEducational Statistics

GURU K MOORTHYGURU K MOORTHYGURU K MOORTHYGURU K MOORTHY

OutlineOutline

Introduction Frequency Distribution Measures of Central Tendency Measures of Dispersion

Outline-ContinuedOutline-Continued

Other Measures Concept of a Population and Sample The Normal Curve Tests for Normality

Learning ObjectivesLearning Objectives

When you have completed this chapter you should be able to:

Know the difference between a variable and an attribute.

Perform mathematical calculations to the correct number of significant figures.

Construct histograms for simple and complex data.

Learning Objectives-cont’d.Learning Objectives-cont’d.

Calculate and effectively use the different measures of central tendency, dispersion, and interrelationship.

Understand the concept of a universe and a sample.

Understand the concept of a normal curve and the relationship to the mean and standard deviation.

Learning Objectives-cont’d.Learning Objectives-cont’d.

Calculate the percent of items below a value, above a value, or between two values for data that are normally distributed.

Calculate the process center given the percent of items below a value

Perform the different tests of normality Construct a scatter diagram and perform

the necessary related calculations.

Definition of Statistics:

1. A collection of quantitative data pertaining to a subject or group. Examples are blood pressure statistics etc.

2. The science that deals with the collection, tabulation, analysis, interpretation, and presentation of quantitative data

IntroductionIntroduction

Two phases of statistics:Descriptive Statistics:

Describes the characteristics of a product or process using information collected on it.

Inferential Statistics (Inductive):Draws conclusions on unknown process

parameters based on information contained in a sample.

Uses probability

IntroductionIntroduction

Types of Data:

Attribute:

Discrete data. Data values can only be integers. Counted data or attribute data. Examples include: How many of the products are

defective? How often are the machines repaired? How many people are absent each day?

Collection of DataCollection of Data

Types of Data:

Attribute:

Discrete data. Data values can only be integers. Counted data or attribute data. Examples include: How many days did it rain last month? What kind of performance was

achieved? Number of defects, defectives

Collection of Data – Cont’d.Collection of Data – Cont’d.

Types of Data:

Variable:

Continuous data. Data values can be any real number. Measured data.

Examples include: How long is each item? How long did it take to complete the

task? What is the weight of the product? Length, volume, time

Significant Figures Rounding

Significant Figures = Measured numbers When you measure something there is

always room for a little bit of error How tall are you 5 ft 9 inches or 5 ft 9.1

inches? Counted numbers and defined numbers ( 12

ins. = 1 ft, there are 6 people in my family)

Significant FiguresSignificant Figures

Significant figures are used to indicate the amount of variation which is allowed in a number.

It is believed to be closer to the actual value than any other digit.

Significant figures:3.69 – 3 significant digits.36.900 – 5 significant digits.

Use Scientific Notation3x10^2 (1 significant digit)3.0x10^2 (2 significant digits)

Significant Figures – Cont’d.Significant Figures – Cont’d.

Rules for Multiplying and Dividing Number of sig. = the same as the number

with the least number of significant digits.6.59 x 2.3 = 1532.65/24 = 1.4 (where 24 is not a

counting number)32.64/24=1.360(24 is a counting

number i.e. 24.00)

Rules for Adding and Subtracting Result can have no more sig. fig. after the

decimal point than the number with the fewest sig. fig. after the decimal point.38.26 – 6 = 32 (6 is not a counting

number)38.2 -6 = 32.2 (6 is a counting number)38.26 – 6.1 = 32.2 (rounded from 32.16) If the last digit >=5 then round up, else

round down

Precision

The precision of a measurement is determined by how reproducible that measurement value is.

For example if a sample is weighed by a student to be 42.58 g, and then measured by another student five different times with the resulting data: 42.09 g, 42.15 g, 42.1 g, 42.16 g, 42.12 g Then the original measurement is not very precise since it cannot be reproduced.

Precision and AccuracyPrecision and Accuracy

Accuracy The accuracy of a measurement is

determined by how close a measured value is to its “true” value.

For example, if a sample is known to weigh 3.182 g, then weighed five different times by a student with the resulting data: 3.200 g, 3.180 g, 3.152 g, 3.168 g, 3.189 g

The most accurate measurement would be 3.180 g, because it is closest to the true “weight” of the sample.

Figure 4-1 Difference between accuracy and precision

Frequency Distribution Measures of Central Tendency Measures of Dispersion

DescribingDescribing DataData

Ungrouped Data Grouped Data

Frequency DistributionFrequency Distribution

2-72-7There are three types of frequency distributions

Categorical frequency distributions Ungrouped frequency distributions Grouped frequency distributions

Frequency DistributionFrequency Distribution

2-72-7Categorical frequency distributions Can be used for data that can be placed

in specific categories, such as nominal- or ordinal-level data.

Examples - political affiliation, religious affiliation, blood type etc.

CategoricalCategorical

2-82-8 Example :Blood Type Frequency Example :Blood Type Frequency DistributionDistribution

Class Frequency Percent

A 5 20

B 7 28

O 9 36

AB 4 16

CategoricalCategorical

2-92-9Ungrouped frequency distributions Ungrouped frequency distributions - can

be used for data that can be enumerated and when the range of values in the data set is not large.

Examples - number of miles your instructors have to travel from home to campus, number of girls in a 4-child family etc.

UngroupedUngrouped

2-102-10 Example :Number of Miles TraveledExample :Number of Miles Traveled

Class Frequency

UngroupedUngrouped

2-112-11 Grouped frequency distributions Can be used when the range of values

in the data set is very large. The data must be grouped into classes that are more than one unit in width.

Examples - the life of boat batteries in hours.

GroupedGrouped

2-122-12Example: Lifetimes of Boat BatteriesExample: Lifetimes of Boat Batteries

Classlimits

ClassBoundaries

Cumulative

24 - 30 23.5 - 37.5 4 4

38 - 51 37.5 - 51.5 14 18

52 - 65 51.5 - 65.5 7 25

frequencyFrequency

GroupedGrouped

Number non conforming

Frequency Relative Frequency

Cumulative Frequency

RelativeFrequency

0 15 0.29 15 0.29

1 20 0.38 35 0.67

2 8 0.15 43 0.83

3 5 0.10 48 0.92

4 3 0.06 51 0.98

5 1 0.02 52 1.00

Table 4-3 Different Frequency Distributions of Data Given in Table 4-1

Frequency DistributionsFrequency Distributions

Frequency Histogram

0 1 2 3 4 5

Number Nonconforming

Frequency HistogramFrequency Histogram

Relative Frequency Histogram

0 1 2 3 4 5

tive F

Relative Frequency Relative Frequency HistogramHistogram

Cumulative Frequency Histogram

0 1 2 3 4 5

Cumulative Frequency Cumulative Frequency HistogramHistogram

The histogram is the most important graphical tool for exploring the shape of data distributions.

Check: http://quarknet.fnal.gov/toolkits/ati/histograms.html for the construction ,analysis and understanding of histograms

The HistogramThe Histogram

The Fast WayStep 1: Find range of distribution, largest - smallest valuesStep 2: Choose number of classes, 5 to 20Step 3: Determine width of classes, one decimal place more than the data, class width = range/number of classesStep 4: Determine class boundariesStep 5: Draw frequency histogram

#classes n

Constructing a HistogramConstructing a Histogram

Number of groups or cellsIf no. of observations < 100 – 5 to 9

cellsBetween 100-500 – 8 to 17 cellsGreater than 500 – 15 to 20 cells

For a more accurate way of drawing a histogram see the section on grouped data in your textbook

Bar Graph Polygon of Data Cumulative Frequency Distribution or

Other Types of Other Types of Frequency Distribution Frequency Distribution

GraphsGraphs

Bar Graph and Polygon of Bar Graph and Polygon of DataData

Cumulative FrequencyCumulative Frequency

Figure 4-6 Characteristics of frequency distributions

Characteristics of FrequencyCharacteristics of FrequencyDistribution GraphsDistribution Graphs

Analysis of HistogramsAnalysis of Histograms

Figure 4-7 Differences due to location, spread, and shape

Analysis of HistogramsAnalysis of Histograms

Figure 4-8 Histogram of Wash Concentration

The three measures in common use are the: Average Median Mode

Measures of Central Measures of Central TendencyTendency

There are three different techniques available for calculating the average three measures in common use are the:

Ungrouped data Grouped data Weighted average

AverageAverage

Average-Ungrouped DataAverage-Ungrouped Data

1 1 2 2

f X f X f X

h = number of cellsh = number of cellsfi=frequencyfi=frequencyXi=midpointXi=midpoint

Average-Grouped DataAverage-Grouped Data

i iiw n

Used when a number of averages are combined with different frequencies

Average-Weighted AverageAverage-Weighted Average

M L if

Lm=lower boundary of the cell with the medianN=total number of observationsCfm=cumulative frequency of all cells below mFm=frequency of median celli=cell interval

Median-Grouped DataMedian-Grouped Data

Boundaries Midpoint Frequency Computation

23.6-26.5 25.0 4 100

26.6-29.5 28.0 36 1008

29.6-32.5 31.0 51 1581

32.6-35.5 34.0 63 2142

35.6-38.5 37.0 58 2146

38.6-41.5 40.0 52 2080

41.6-44.5 43.0 34 1462

44.6-47.5 46.0 16 736

47.6-50.5 49.0 6 294

Total 320 11549

Table 4-7 Frequency Distribution of the Life of 320 tires in 1000 km

Example ProblemExample Problem

M L if

320154

235.6 3 35.958

Median-Grouped DataMedian-Grouped Data

Using data from Table 4-7

ModeMode

The Mode is the value that occurs with the greatest frequency.

It is possible to have no modes in a series or numbers or to have more than one mode.

Figure 4-9 Relationship among average, median and mode

Relationship Among theRelationship Among theMeasures of Central Measures of Central

TendencyTendency

Range Standard Deviation Variance

Measures of DispersionMeasures of Dispersion

The range is the simplest and easiest to calculate of the measures of dispersion.

Range = R = Xh - Xl Largest value - Smallest value in

data set

MeasuresMeasures of Dispersion- of Dispersion-RangeRange

Sample Standard Deviation:

Xi Xi n

Measures of Dispersion-Measures of Dispersion-Standard DeviationStandard Deviation

Ungrouped Technique

1 1( )

i in Xi Xi

Standard DeviationStandard Deviation

( ) ( )

i i i iii

n f X f Xs

Standard DeviationStandard Deviation

Grouped Technique

Relationship Between the Relationship Between the Measures of DispersionMeasures of Dispersion

As n increases, accuracy of R decreases Use R when there is small amount of data

or data is too scattered If n> 10 use standard deviation A smaller standard deviation means better

quality

Relationship Between the Relationship Between the Measures of DispersionMeasures of Dispersion

Figure 4-10 Comparison of two distributions with equal average and range

Other MeasuresOther Measures

There are three other measures that are frequently used to analyze a collection of data: Skewness Kurtosis Coefficient of Variation

Skewness is the lack of symmetry of the data.

For grouped data:3

( ) /h

i iif X X n

SkewnessSkewness

Figure 4-11 Left (negative) and right (positive) skewness distributions

Kurtosis provides information regrading the shape of the population distribution (the peakedness or heaviness of the tails of a distribution).

For grouped data:4

( ) /h

i iif X X n

KurtosisKurtosis

Figure 4-11 Leptokurtic and Platykurtic distributions

Correlation variation (CV) is a measure of how much variation exists in relation to the mean.

Coefficient of VariationCoefficient of Variation

(100%)sCV

Population Set of all items that possess a characteristic of interest

Sample Subset of a population

Population and SamplePopulation and Sample

Parameter is a characteristic of a population, i.o.w. it describes a population Example: average weight of the population, e.g. 50,000 cans made in a month.Statistic is a characteristic of a sample, used to make inferences on the population parameters that are typically unknown, called an estimator Example: average weight of a sample of 500 cans from that month’s output, an estimate of the average weight of the 50,000 cans.

Parameter and StatisticParameter and Statistic

Characteristics of the normal curve: It is symmetrical -- Half the cases are to

one side of the center; the other half is on the other side.

The distribution is single peaked, not bimodal or multi-modal

Also known as the Gaussian distribution

The Normal CurveThe Normal Curve

Characteristics:

Most of the cases will fall in the center portion of the curve and as values of the variable become more extreme they become less frequent, with "outliers" at the "tail" of the distribution few in number. It is one of many frequency distributions.

The Normal CurveThe Normal Curve

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Normal distributions can be transformed to standard normal distributions by the formula:

Standard Normal DistributionStandard Normal Distribution

Relationship between the Relationship between the

Mean and Standard Mean and Standard DeviationDeviation

Same mean but different standard deviation

IF THE DISTRIBUTION IS NORMAL

Then the mean is the best measure of central tendencyMost scores “bunched up” in

middleExtreme scores are less frequent,

therefore less probable

Normal DistributionNormal Distribution

Percent of items included between certain values of the std. deviation

Normal DistributionNormal Distribution

Histogram Skewness Kurtosis

Tests for NormalityTests for Normality

Histogram:

ShapeSymmetrical

The larger the sampler size, the better the judgment of normality. A minimum sample size of 50 is recommended

Skewness (a3) and Kurtosis (a4)” Skewed to the left or to the right (a3=0 for

a normal distribution) The data are peaked as the normal

distribution (a4=3 for a normal distribution)

The larger the sample size, the better the judgment of normality (sample size of 100 is recommended)

Probability Plots Order the data from the smallest to the

largest Rank the observations (starting from 1 for

the lowest observation) Calculate the plotting position

100( 0.5)iPP

Where i = rank PP=plotting position n=sample size

Procedure: Order the data Rank the observations Calculate the plotting position

Probability PlotsProbability Plots

Procedure cont’d: Label the data scale Plot the points Attempt to fit by eye a “best

line” Determine normality

Procedure cont’d: Order the data Rank the observations Calculate the plotting position Label the data scale Plot the points Attempt to fit by eye a “best line” Determine normality

Chi-Square Test

2 Chi-squared

Observed value in a cell

Expected value for a cell

Chi-Square Goodness of Fit Chi-Square Goodness of Fit TestTest

The simplest way to determine if a The simplest way to determine if a cause and-effect relationship exists cause and-effect relationship exists between two variablesbetween two variables

Scatter DiagramScatter Diagram

Figure 4-19 Scatter Diagram

Supplies the data to confirm a Supplies the data to confirm a hypothesis that two variables are hypothesis that two variables are relatedrelated

Provides both a visual and statistical Provides both a visual and statistical means to test the strength of a means to test the strength of a relationshiprelationship

Provides a good follow-up to cause and Provides a good follow-up to cause and effect diagramseffect diagrams

Scatter DiagramScatter Diagram

Straight Line FitStraight Line Fit

[( )( ) /

[( ) / ]

/ ( / )

xy x y nm

a y n m x n

y a mx

Where m=slope of the line and a is the intercept on the y axis

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