StatisticsStatistics

Measures of Central Measures of Central TendencyTendency

Why Describe Central Tendency?Why Describe Central Tendency?

• Data often cluster around a central value that Data often cluster around a central value that lies between the two extremes. This single lies between the two extremes. This single number can describe the value of scores in number can describe the value of scores in the entire data set.the entire data set.

• There are three measures of central There are three measures of central tendency.tendency.

1) Mean1) Mean

2) Median2) Median

3) Mode 3) Mode

Measures of central tendency are scores Measures of central tendency are scores that represent the center of the that represent the center of the distribution.distribution.

Three of the most common measures of Three of the most common measures of central tendency are:central tendency are:– MeanMean– MedianMedian– ModeMode

MeanMean

The most commonly used measure of central The most commonly used measure of central tendencytendency

When people ask about the “average” of a group of When people ask about the “average” of a group of scores, they usually are referring to the mean.scores, they usually are referring to the mean.

The mean is the sum of all the scores in the The mean is the sum of all the scores in the distribution divided by the total scores (the distribution divided by the total scores (the mathematical average). mathematical average).

Mean (con’t)Mean (con’t)

Mean of a sampleMean of a sample

Mean of a populationMean of a population

Mean (con’t)Mean (con’t)

7575 8282 7272 6868 8989

9191 7878 9494 8888 7575

Exam Scores

sum all scoresX n = total number of scores for the sample

Formula 7.AFormula 7.A

Arithmetic Mean

Mean - Ungrouped Data

For a population:

N

X

N

XXXX N

...321

For a sample:

n

X

n

XXXXX n

...321

Mean - Grouped Data

For a population:

N

fX

For a sample:

n

fXX

Formula 7.BFormula 7.B

7.7

Mean (con’t)Mean (con’t)

Performance and Memory Study

10.59.58.57.56.55.54.53.52.51.50

1

2

3

4

5

6

Number of Words Recalled

Fre

quen

cy

22 22 33 33 44

44 44 44 44 1100

The mean includes the weight of every score.

Table 7.2Table 7.2

Arithmetic Mean

Table 7.2 | Approximation of the Arithmetic Mean from a Frequency Distribution

Class(net profit in

millions of rupees)

Absolute ClassFrequency (number

of companies inclass) f

ClassMidpoint X fX

-1,250 to under 0 0 to under 1,250

1,250 to under 2,5002,500 to under 3,7503,750 to under 5,0005,000 to under 6,2506,250 to under 7,5007,500 to under 8,750

8,750 to under10,000

6491815 3 2 4 2

1f = N = 100

-625 625

1,875 3,125 4,375 5,625 6,875 8,125 9,375

-3,750 30,625 33,750 46,875 13,125 11,250 27,500

16,2509,375

fX =185,000

Estimated arithmetic mean = Rs.1,850 (based on the ratio 185,000/100)

7.9

Pros and cons of using meanPros and cons of using mean

ProsPros– Summarizes data in a way that is easy to understand.Summarizes data in a way that is easy to understand.– Uses all the data Uses all the data – Used in many statistical applicationsUsed in many statistical applications

Cons Cons – Affected by extreme valuesAffected by extreme values

E.g., average salary at a companyE.g., average salary at a company– 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000;

12,000; 12,000; 12,000; 12,000; 20,000; 390,00012,000; 12,000; 12,000; 12,000; 20,000; 390,000– Mean = $44,167Mean = $44,167

Merits and demerits of meanMerits and demerits of mean MeritsMerits

– Mean is well understood by most peopleMean is well understood by most people– Computation of mean is easyComputation of mean is easy

DemeritsDemerits– Sensitive to extreme valueSensitive to extreme value– For example:For example:

X={1,1,1,1,2,9}, mean(X)=2.5 which does not X={1,1,1,1,2,9}, mean(X)=2.5 which does not reflect the actually central tendency of this set of reflect the actually central tendency of this set of numbersnumbers

MedianMedian

The middle score of the distribution when all the The middle score of the distribution when all the scores have been ranked.scores have been ranked.

If there are an even number of scores, the median If there are an even number of scores, the median is the average of the two middle scores.is the average of the two middle scores.

MedianMedian DefinitionDefinition

– It divides the numbers into two halves such It divides the numbers into two halves such that the number of items below it is the same that the number of items below it is the same as the number of items above itas the number of items above it

– Suppose we have n numbers xSuppose we have n numbers x11, x, x22, ……, x, ……, xnn. .

– Median is defined asMedian is defined as

Median (con’t)Median (con’t)

22

22

33

33

44

44

44

44

44

1010

Number of Words Recalled in Performance Study

Formula 7.CFormula 7.C

The Median

Median - Ungrouped Data

For a population:

For a sample:

21 NX

21 nXm

X = population (or sample) valueN = number of observations in populationn = number of observations in samplesubscript = position of X in ordered array

Median - Grouped Data

For a population:

For a sample:

wf

FNL

)2/(

wf

FnLm

)2/(

L = the median class’s lower limitf = its absolute frequencyw = its widthF = the sum of frequencies up to (but not including) those of the median class

Formula 7.DFormula 7.D

7.15

Merits and demerits of medianMerits and demerits of median MeritsMerits

– Another widely used measure of central Another widely used measure of central tendencytendency

– It is not influenced by extreme valuesIt is not influenced by extreme values

DemeritsDemerits– When the number of items are small, median When the number of items are small, median

may not be representative, because it is a may not be representative, because it is a positional averagepositional average

ModeMode The most frequent score in the distribution.The most frequent score in the distribution.

A distribution where a single score is most frequent has one A distribution where a single score is most frequent has one mode and is called mode and is called unimodal.unimodal.

A distribution that consists of only one of each score has A distribution that consists of only one of each score has n n modesmodes..

When there are ties for the most frequent score, the When there are ties for the most frequent score, the distribution is distribution is bimodalbimodal if two scores tie or if two scores tie or multimodalmultimodal if more if more than two scores tie.than two scores tie.

Mode (con’t)Mode (con’t)

22

22

33

33

44

44

44

44

44

1010

Number of Words Recalled in Performance Study

The mode is 4.

Mode (con’t)Mode (con’t)

7272 7272 7373 7676 7878

8181 8383 8585 8585 8686

8787 8888 9090 9191 9292

This distribution is bimodal.

Demonstration

CalculationsCalculations Key: dependent measure is reaction timeKey: dependent measure is reaction time

– Time it takes to say the colorTime it takes to say the color

Determine the mean, median, and mode Determine the mean, median, and mode of the datasets in the handout.of the datasets in the handout.

Shape of the DistributionShape of the Distribution

Skew refers to the general shape of a distribution when it is graphed.

Symmetrical = zero skew

Scores clustered on the high or low end of a distribution = skewed distribution

Symmetrical Distribution

24.5 29.5 34.5 39.5 44.5 49.5 54.5 59.5 64.5 69.50

2

4

6

8

10

12

14

16

Scores

Freq

uenc

y

The mean, median, and mode are the same.

The normal distribution is the “ideal” The normal distribution is the “ideal” symmetrical distributionsymmetrical distribution

Distributions that are skewed have one Distributions that are skewed have one side of the distribution where the data side of the distribution where the data

frequency tapers offfrequency tapers off

Skewed DistributionSkewed Distribution

Positive Skew

27 32 37 42 47 52 57 62 67 72 770

2

4

6

8

10

12

Scores

Freq

uenc

y

Tail points in the positive direction.

Skewed DistributionSkewed Distribution

Negative Skew

27 32 37 42 47 52 57 62 67 72 770

2

4

6

8

10

12

Scores

Fre

qu

en

cy

Tail points in negative direction.

The mean will either underestimate or The mean will either underestimate or overestimate the center of skewed overestimate the center of skewed

distributions.distributions. Positive Skew

27 32 37 42 47 52 57 62 67 72 770

2

4

6

8

10

12

Scores

Freq

uen

cy

Negative Skew

27 32 37 42 47 52 57 62 67 72 770

2

4

6

8

10

12

Scores

Fre

qu

en

cy

Mode

MedianMean

Mode

MeanMedian

KurtosisKurtosis

Measure of the flatness or peakedness of Measure of the flatness or peakedness of the distributionthe distribution

Measures of LocationMeasures of Location

Measures that are allied to the median include Measures that are allied to the median include the quartiles, deciles and percentiles, because the quartiles, deciles and percentiles, because they are also based on their position in a series they are also based on their position in a series of observations. of observations.

These measures are referred to as measures of These measures are referred to as measures of location and not the measures of central location and not the measures of central tendency as they describe the position of one tendency as they describe the position of one score relative to the others rather than the whole score relative to the others rather than the whole set of data.set of data.

Measures of LocationMeasures of Location

QuartileQuartile– One fourth (1/4)One fourth (1/4)

First (1/4), Second (1/2), Third (3/4) First (1/4), Second (1/2), Third (3/4)

DecileDecile– One tenth (1/10)One tenth (1/10)

10%, 20%, …90%10%, 20%, …90%

Percentile Percentile – One of hundreds (1/100)One of hundreds (1/100)

1%, 2%, ….99%1%, 2%, ….99%

QuartilesQuartiles

The median divides the data into two equal sets.

The lower quartile is the value of the middle of the first set, where 25% of the values are smaller than Q1 and 75% are larger. This first quartile takes the notation Q1.

The upper quartile is the value of the middle of the second set, where 75% of the values are smaller than Q3 and 25% are larger. This third quartile takes the notation Q3.

QuartilesQuartiles

Example 1 – Upper and lower quartiles Data 6, 47, 49, 15, 43, 41, 7, 39, 43, 41, 36 Ordered data 6, 7, 15, 36, 39, 41, 41, 43, 43, 47, 49 Median 41 Upper quartile 43 Lower quartile 15

QuartileQuartile

LL0 0 = Lower limit of the = Lower limit of the ii-th Quartile class-th Quartile class nn = Total number of observations in the = Total number of observations in the distributiondistribution h = Class width of the h = Class width of the ii-th Quartile class-th Quartile class ffii = Frequency of the = Frequency of the ii-th Quartile class-th Quartile class F = Cumulative frequency of the class prior to F = Cumulative frequency of the class prior to the the ii-th quartile class-th quartile class

3,2,1,4

iFn

i

f

hLQ

iii

DecileDecile

LL0 0 = Lower limit of the = Lower limit of the ii-th Decile class-th Decile class nn = Total number of observations in the = Total number of observations in the distributiondistribution h = Class width of the h = Class width of the ii-th Decile class-th Decile class ffii = Frequency of the = Frequency of the ii-th Decile class-th Decile class F = Cumulative frequency of the class prior to F = Cumulative frequency of the class prior to the the ii-th Decile class-th Decile class

3,2,1,10

iFn

i

f

hLP

iii

PercentilePercentile

LL0 0 = Lower limit of the = Lower limit of the ii-th Percentile class-th Percentile class nn = Total number of observations in the = Total number of observations in the distributiondistribution h = Class width of the h = Class width of the ii-th Percentile class-th Percentile class ffii = Frequency of the = Frequency of the ii-th Percentile class-th Percentile class F = Cumulative frequency of the class prior to F = Cumulative frequency of the class prior to the the ii-th Percentile class-th Percentile class

3,2,1,100

iFn

i

f

hLP

iii

Example-1: Percentile of Ungroup Example-1: Percentile of Ungroup datadata

Consider the observations 11, 14, 17, 23, Consider the observations 11, 14, 17, 23, 27, 32, 40, 49, 54, 59, 71 and 80. To 27, 32, 40, 49, 54, 59, 71 and 80. To determine the 29th percentile? determine the 29th percentile?

we note that which is not an we note that which is not an

integer. Thus the next higher integer 4 here integer. Thus the next higher integer 4 here will determine the 29th percentile value. will determine the 29th percentile value. On inspection POn inspection P2929 = 23 = 23

,48.3)1229(100

1

Example-2: Find 3Example-2: Find 3rdrd Quartiles, 1 Quartiles, 1stst Decile and 29Decile and 29thth Percentile Percentile

Marks Marks

No. of students No. of students Cumulative Cumulative frequenciesfrequencies

40-5040-50 66 66

50-6050-60 1111 1717

60-7060-70 1919 3636

70-8070-80 1717 5353

80-9080-90 1313 6666

90-10090-100 44 7070

TotalTotal 7070

Example-3: Find 30Example-3: Find 30thth percentile percentile

Age in yearsAge in years Number of birthsNumber of births Cumulative number Cumulative number of birthsof births

14.5-19.514.5-19.5 677677 677677

19.5-24.519.5-24.5 19081908 25852585

24.5-29.524.5-29.5 17371737 43324332

29.5-34.529.5-34.5 10401040 53625362

34.5-39.534.5-39.5 294294 56565656

39.5-44.539.5-44.5 9191 57475747

44.5-49.544.5-49.5 1616 57635763

All agesAll ages 57635763 --

Determine Percentile ClassDetermine Percentile Class

First determine the percentile class.First determine the percentile class. If, N =5736, and we have to find 30If, N =5736, and we have to find 30thth

percentile, then percentile class will be the percentile, then percentile class will be the class which has cumulative frequency class which has cumulative frequency below: below:

.9.17285763)100/30(100

ni

3030thth Percentile class Percentile class

Age in yearsAge in years Number of birthsNumber of births Cumulative number Cumulative number of birthsof births

14.5-19.514.5-19.5 677677 677677

19.5-24.519.5-24.5 19081908 25852585

24.5-29.524.5-29.5 17371737 43324332

29.5-34.529.5-34.5 10401040 53625362

34.5-39.534.5-39.5 294294 56565656

39.5-44.539.5-44.5 9191 57475747

44.5-49.544.5-49.5 1616 57635763

All agesAll ages 57635763 --

1728.9

Percentiles and LocationPercentiles and LocationTop of 1st Q-tile

Top of 2nd Q-tile (med)

Top of 3rd

Q-tile

25th percentile 50th percentile 75th percentile

If you’re at the 75th percentile, or the 3rd quartile, of test scores this means 75% of other test takers scored below you.

Which is a better score, 1st percentile or 99th percentile?

Box PlotsBox Plots

Five pieces of data are Five pieces of data are needed to construct a box needed to construct a box plot: plot:

the the Minimum ValueMinimum Value,, the the First QuartileFirst Quartile,, the the MedianMedian,, the the Third QuartileThird Quartile, and, and the the Maximum ValueMaximum Value..

A box plot is a graphical display, based on quartiles, that helps to picture a set of

data.

Example 4Example 4

Based on a sample of 20 deliveries,

Buddy’s Pizza determined the following information. The

minimum delivery time was 13 minutes and the maximum 30

minutes. The first quartile was 15 minutes, the median 18

minutes, and the third quartile 22 minutes. Develop a box plot

for the delivery times.

Example 4 Example 4 continuedcontinued