conducting descriptive statistics dr. k. a. korb university of jos

Conducting Descriptive Conducting Descriptive StatisticsStatistics

Dr. K. A. KorbDr. K. A. Korb

University of JosUniversity of Jos

OutlineOutline

FrequencyFrequency Measures of Central TendencyMeasures of Central Tendency

ModeMode MeanMean MedianMedian

Measures of VariabilityMeasures of Variability RangeRange VarianceVariance Standard DeviationStandard Deviation

Dr. K. A. KorbUniversity of Jos

FrequencyFrequency Frequency: Number of times a score occurs.Frequency: Number of times a score occurs. Frequencies can either be reported by a table or by a chart.Frequencies can either be reported by a table or by a chart. Reporting frequency is typically only informative for nominal (categorical) Reporting frequency is typically only informative for nominal (categorical)

datadata

Menu Availability Menu Availability Frequency TableFrequency Table

FoodFood FrequencFrequencyy

TuwoTuwo 77

Moi MoiMoi Moi 55

Jollof Jollof RiceRice

88

Food is ReadyFood is Ready Menu MenuMondayMonday TuesdayTuesday WednesdWednesd

ayayThursdayThursday FridayFriday

Week 1Week 1 Moi-MoiMoi-Moi Moi-MoiMoi-Moi Jollof RiceJollof Rice Jollof RiceJollof Rice TuwoTuwo

Week 2Week 2 TuwoTuwo Jollof RiceJollof Rice TuwoTuwo Moi-MoiMoi-Moi Jollof RiceJollof Rice

Week 3Week 3 Moi-MoiMoi-Moi TuwoTuwo Jollof RiceJollof Rice Jollof RiceJollof Rice TuwoTuwo

Week 4Week 4 Jollof RiceJollof Rice Moi-MoiMoi-Moi TuwoTuwo TuwoTuwo Jollof RiceJollof Rice


Frequency Pie ChartFrequency Pie ChartPercentage of Availability of Menu Options at

Food is Ready

Tuwo35%

Jollof Rice40%

Moi-Moi25%


Masters Degree Masters Degree Frequency TableFrequency Table

UniversitUniversityy

FrequencFrequencyy

UniJosUniJos 77

ABUABU 22

IbadanIbadan 11

S/NS/N UniUni

11 UniJosUniJos

22 ABUABU

33 UniJosUniJos

44 UniJosUniJos

55 UniJosUniJos

66 UniJosUniJos

77 UniJosUniJos

88 ABUABU

99 UniJosUniJos

1010 IbadanIbadan

Where did UniJos PhD students Where did UniJos PhD students earn their Masters Degree?earn their Masters Degree?


Frequency Bar ChartFrequency Bar Chart

University where PhD students read their Masters Degree

012345678

UniJos ABU Ibadan

University

Freq

uenc

y

.


Types of StatisticsTypes of Statistics Three fundamental types of statisticsThree fundamental types of statistics

1.1. DescriptiveDescriptive: Explains trends in your sample: Explains trends in your sample Central TendencyCentral Tendency

ModeMode MeanMean MedianMedian

VariabilityVariability Range Range Standard DeviationStandard Deviation

FrequenciesFrequencies2.2. Significance of MeansSignificance of Means: Determines whether differences : Determines whether differences

between groups of individuals are large enough to be between groups of individuals are large enough to be meaningfulmeaningful

t-tests, ANOVA, ANCOVAt-tests, ANOVA, ANCOVA3.3. Relationship between VariablesRelationship between Variables: Compares the : Compares the

relationship between multiple variables within the same relationship between multiple variables within the same sample of individualssample of individuals

CorrelationCorrelation RegressionRegression


Descriptive StatisticsDescriptive Statistics June received a score of 20 on the Extraversion June received a score of 20 on the Extraversion

Personality Questionnaire.Personality Questionnaire. With just this information, we cannot interpret her score.With just this information, we cannot interpret her score.

What does the average person score on the Questionnaire?What does the average person score on the Questionnaire? What is the range of typical scores on the Questionnaire?What is the range of typical scores on the Questionnaire?

What two things do you need to know to interpret What two things do you need to know to interpret her score?her score? Average: Typical scoreAverage: Typical score

Average: 30Average: 30 Range of scoresRange of scores

Standard Deviation: 10Standard Deviation: 10 Now we can say two things:Now we can say two things:

June has less Extraversion than the typical person June has less Extraversion than the typical person because her score of 20 was less than the average score because her score of 20 was less than the average score of 30.of 30.

June is one standard deviation below the mean (30 – 20 June is one standard deviation below the mean (30 – 20 = 10, the standard deviation), so she has considerably = 10, the standard deviation), so she has considerably less extraversion than most people.less extraversion than most people.


Central TendencyCentral Tendency

Average: Typical performanceAverage: Typical performance Mode: Most frequent scoreMode: Most frequent score Mean: Sum of scores divided by number Mean: Sum of scores divided by number

of scoresof scores Median: Middle score in the distribution Median: Middle score in the distribution

The next slides give an example of The next slides give an example of the the Mode.Mode.


What is the typical meal served at What is the typical meal served at the the Food is Ready?Food is Ready?

Food is ReadyFood is Ready Menu MenuMondayMonday TuesdayTuesday WednesdWednesd

ayayThursdayThursday FridayFriday

Week 1Week 1 Moi-MoiMoi-Moi Moi-MoiMoi-Moi Jollof RiceJollof Rice Jollof RiceJollof Rice TuwoTuwo

Week 2Week 2 TuwoTuwo Jollof RiceJollof Rice TuwoTuwo Moi-MoiMoi-Moi Jollof RiceJollof Rice

Week 3Week 3 Moi-MoiMoi-Moi TuwoTuwo Jollof RiceJollof Rice Jollof RiceJollof Rice TuwoTuwo

Week 4Week 4 Jollof RiceJollof Rice Moi-MoiMoi-Moi TuwoTuwo TuwoTuwo Jollof RiceJollof Rice

FrequencyFrequency Tuwo: 7Tuwo: 7 Moi-Moi: 5Moi-Moi: 5 Jollof Rice: 8Jollof Rice: 8

Jollof Rice is on the menu Jollof Rice is on the menu the most frequently, so the most frequently, so the the Mode Mode is is Jollof Rice.Jollof Rice.


CentralCentral Tendency Tendency

Average: Typical performanceAverage: Typical performance Mode: Most frequent scoreMode: Most frequent score

Best for nominal (categorical) dataBest for nominal (categorical) data Represent by a pie graphRepresent by a pie graph

Mean: Sum of scores divided by number Mean: Sum of scores divided by number of scoresof scores

Median: Middle score in the distributionMedian: Middle score in the distribution

The next slides give explain the The next slides give explain the Mean.Mean.


What is the typical price of oranges What is the typical price of oranges at the market?at the market?CosCos

tt

5050

5050

5555

6060

6060

6565

7070

7070

7070

7070

50+50+55+60+60+65+70+70+70+7010

= 62010

= 62

Mean

We can also find the Mode, the most frequent score.

Mode = 70

Mean = 62


What is the typical price of oranges What is the typical price of oranges at the market?at the market?CosCos

tt5050

5050

5555

6060

6060

6565

7070

7070

7070

7070

150150

50+50+55+60+60+65+70+70+70+70+15011

= 77011

= 70

Mean

Mean = 70

The next person who goes to the market is a bature and they get charged N150 for the oranges. Let’s

recalculate.

The mean has jumped from 62 to 70 with just one additional data point.


Price of OrangesPrice of Oranges

Cost of Oranges

0

1

2

3

4

50 60 70 80 90 100 110 120 130 140 150

Cost

Fre

quen

cy

.

This frequency chart clearly shows that the N150 purchase is an outlier – a data point that is far from the

other data points.Dr. K. A. KorbUniversity of Jos

Price of OrangesPrice of OrangesCosCos

tt

5050

5050

5555

6060

6060

6565

7070

7070

7070

7070

Median

50 50 55 60 60 65 70 70 70 70

Median = 62.5

The Median is the middle score. First calculate the median for the data without the bature purchase.

When arranged from smallest to largest, there are 5 data points to the left and 5 data points to the right. To find the median with an even set of data points, calculate the mean of the two middle numbers – 60

and 65.


Price of OrangesPrice of Oranges

Median

50 50 55 60 60 65 70 70 70 70 150

Median = 65

Now let’s calculate the median with the bature purchase.

When arranged from smallest to largest, there are 5 data points to the left and 5 data points to the right

of the number 65. The median with an odd set of data points is simply the middle number.

CosCostt

5050

5050

5555

6060

6060

6565

7070

7070

7070

7070

150150Dr. K. A. KorbUniversity of Jos


Without Bature Without Bature PurchasePurchase

With Bature With Bature PurchasePurchase

MeanMean 6262 7070

MedianMedian 62.562.5 6565

The Mean changed substantially with the The Mean changed substantially with the outlier bature purchase.outlier bature purchase.

The median was not strongly influenced by The median was not strongly influenced by the outlying score.the outlying score.



Average: Typical performanceAverage: Typical performance Mode: Most frequent scoreMode: Most frequent score

Best for categorical dataBest for categorical data Represent by a pie graphRepresent by a pie graph

Mean: Sum of scores divided by number of Mean: Sum of scores divided by number of scoresscores

Most mathematically defendableMost mathematically defendable Calculate and report for virtually all numerical dataCalculate and report for virtually all numerical data Affected by skewAffected by skew

Median: Middle score in the distribution Median: Middle score in the distribution Best for skewed dataBest for skewed data


VariabilityVariability

Variability: The spread of scoresVariability: The spread of scores Range: Highest and lowest scores in the Range: Highest and lowest scores in the

distributiondistribution Variance: Mathematical degree of Variance: Mathematical degree of

spreadspread Standard Deviation: Mathematical index Standard Deviation: Mathematical index

of spread in original measurement unitsof spread in original measurement units

The next slides give an example of The next slides give an example of the the Range.Range.



Range: Report highest and lowest valuesRange: Report highest and lowest values The number of children ranged from 0 to 13.The number of children ranged from 0 to 13.

Number of Children per Family

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Number of Children

Per

cent

age

.




distributiondistribution Simply gives readers an idea of the spread of scores.Simply gives readers an idea of the spread of scores. Not mathematically useful for calculating statistics.Not mathematically useful for calculating statistics.

Variance: Mathematical degree of spreadVariance: Mathematical degree of spread Standard Deviation: Mathematical index of Standard Deviation: Mathematical index of

spread in original measurement unitsspread in original measurement units

The next slides give an example of the The next slides give an example of the Variance.Variance.


VariabilityVariabilityTo Calculate the VariabilityTo Calculate the Variability1.1. Subtract all scores from the Subtract all scores from the

mean (deviation)mean (deviation) If we summed up the If we summed up the

deviation scores, they will deviation scores, they will always always add up to 0 because of add up to 0 because of the mathematical properties the mathematical properties of the mean.of the mean.

To solve this problem, we To solve this problem, we square the deviation.square the deviation.

2.2. Square the deviationSquare the deviation3.3. Sum the deviationSum the deviation22

4.4. Divide by total number of scoresDivide by total number of scores 4.54.5

Since we are summing up the Since we are summing up the deviation and dividing by the deviation and dividing by the total number of scores, the total number of scores, the variance is effectively the variance is effectively the average squared deviation of average squared deviation of each score from the mean.each score from the mean.

Mean = 10

Scores on a 15 point Scores on a 15 point Continuous AssessmentContinuous Assessment

ScoreScore DeviationDeviation DeviationDeviation22

1111 11 11

88 -2-2 44

77 -3-3 99

1313 33 99

1212 22 44

99 -1-1 11

88 -2-2 44

1212 22 44

SumSum 00 3636

Divide by 8Divide by 8 4.54.5Dr. K. A. KorbUniversity of Jos

VariabilityVariability Variability: The spread of scoresVariability: The spread of scores

Range: Highest and lowest scores in the Range: Highest and lowest scores in the distributiondistribution

Simply gives readers an idea of the spread of scores.Simply gives readers an idea of the spread of scores. Not mathematically useful for calculating statistics.Not mathematically useful for calculating statistics.

Variance: Mathematical degree of spreadVariance: Mathematical degree of spread The variance is difficult to interpret because it is in The variance is difficult to interpret because it is in

squared units of the original data points.squared units of the original data points. Standard Deviation: Mathematical index of Standard Deviation: Mathematical index of

spread in original measurement unitsspread in original measurement units

The next slides give an example of the The next slides give an example of the Standard Deviation.Standard Deviation.


VariabilityVariabilityScores on a 15 point Scores on a 15 point

Continuous AssessmentContinuous Assessment

ScoreScore DeviationDeviation DeviationDeviation22

1111 11 11

88 -2-2 44

77 -3-3 99

1313 33 99

1212 22 44

99 -1-1 11

88 -2-2 44

1212 22 44

SumSum 00 3636

Divide by 8Divide by 8 4.54.5

Square RootSquare Root 2.122.12

To Calculate the To Calculate the Standard Standard DeviationDeviation

Calculate the varianceCalculate the variance Take the square root of Take the square root of

the variancethe variance

By taking the square By taking the square root of the variance, the root of the variance, the Standard Deviation (SD) Standard Deviation (SD) is back in the original is back in the original units.units.

A SD of 2.12 means that A SD of 2.12 means that the typical person the typical person deviates from the mean deviates from the mean score of 10 by about score of 10 by about 2.12 points.2.12 points. Dr. K. A. Korb

University of Jos



distributiondistribution Simply gives readers an idea of the spread of scores.Simply gives readers an idea of the spread of scores. Not mathematically useful for calculating statistics.Not mathematically useful for calculating statistics.

Variance: Mathematical degree of spreadVariance: Mathematical degree of spread The variance is difficult to interpret because it is in The variance is difficult to interpret because it is in

squared units of the original data points.squared units of the original data points. Standard Deviation: Mathematical index of Standard Deviation: Mathematical index of

variability in original measurement unitsvariability in original measurement units The Standard Deviation can be interpreted in the The Standard Deviation can be interpreted in the

original scale.original scale. The Standard Deviation is used in many statistical The Standard Deviation is used in many statistical

procedures.procedures.


conducting descriptive statistics dr. k. a. korb university of jos

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