Introduction to Probability Introduction to Probability and Statisticsand Statistics

Thirteenth Edition Thirteenth Edition

Chapter 2

Describing Data

with Numerical Measures

Describing Data with Numerical Measures

• Graphical methods may not always be sufficient for describing data.

• Numerical measures can be created for both populations populations and samples.samples.

– A parameter parameter is a numerical descriptive measure calculated for a populationpopulation.

– A statisticstatistic is a numerical descriptive measure calculated for a samplesample.

Central Tendency (Center) and Dispersion (Variability)

A distribution is an ordered set of numbers showing

how many times each occurred, from the lowest to

the highest number or the reverse

Central tendency: measures of the degree to

which scores are clustered around the mean of a

distribution

Dispersion: measures the fluctuations (variability)

around the characteristics of central tendency

1. Measures of Center

• A measure along the horizontal axis of the data distribution that locates the center center of the distribution.

Arithmetic Mean or Average

• The meanmean of a set of measurements is the sum of the measurements divided by the total number of measurements.

where n = number of measurements

sum of all the measurementsxi

n

xx

n

ii

1

Example

•The set: 2, 9, 1, 5, 6

n

xx i 6.6

5

33

5

651192

If we were able to enumerate the whole population, the population meanpopulation mean would be called (the Greek letter “mu”).

• The medianmedian of a set of measurements is the middle measurement when the measurements are ranked from smallest to largest.

• The position of the medianposition of the median is

Median

0.5(n + 1)

once the measurements have been ordered.

• The set : 2, 4, 9, 8, 6, 5, 3 n = 7• Sort : 2, 3, 4, 5, 6, 8, 9• Position: .5(n + 1) = .5(7 + 1) = 4th

Median = 4th largest measurement

• The set: 2, 4, 9, 8, 6, 5 n = 6

• Sort: 2, 4, 5, 6, 8, 9

• Position: .5(n + 1) = .5(6 + 1) = 3.5th

Median = (5 + 6)/2 = 5.5 — average of the 3rd and 4th measurements

Example

Mode• The modemode is the measurement which occurs

most frequently.

• The set: 2, 4, 9, 8, 8, 5, 3

• The mode is 88, which occurs twice

• The set: 2, 2, 9, 8, 8, 5, 3

• There are two modes—88 and 22 (bimodalbimodal)

• The set: 2, 4, 9, 8, 5, 3

• There is no modeno mode (each value is unique).

• Mean?

• Median?

• Mode? (Highest peak)

The number of quarts of milk purchased by 25 households:

0 0 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5

2.225

55

n

xx i

2m

2mode Quarts

Rela

tive fre

quency

543210

10/25

8/25

6/25

4/25

2/25

0

Example

Extreme Values• The mean is more easily affected by extremely

large or small values than the median.

•The median is often used as a measure of center when the distribution is skewed.

Skewed left: Mean < Median

Skewed right: Mean > Median

Symmetric: Mean = Median

Extreme Values

2. Measures of Variability• A measure along the horizontal axis of the data distribution

that describes the spread spread of the distribution from the center.

RangeDifference between maximum and minimum values

Interquartile RangeDifference between third and first quartile (Q3 - Q1)

VarianceAverage*of the squared deviations from the mean

Standard DeviationSquare root of the variance

Definitions of population variance and sample variance differ slightly.

The Range• The range, R,range, R, of a set of n measurements is the difference between the largest and smallest measurements.

• Example: Example: A botanist records the number of petals on 5 flowers:

5, 12, 6, 8, 14• The range is R = 14 – 5 = 9.R = 14 – 5 = 9.

Quick and easy, but only uses 2 of the 5 measurements.Quick and easy, but only uses 2 of the 5 measurements.

• The lower and upper quartiles (Qlower and upper quartiles (Q1 1 and Qand Q33), ), can be calculated as follows:

• The position of Qposition of Q11 is

Interquartiles Range (IQR= Q Q1 1 – Q– Q33)

0.75(n + 1)

0.25(n + 1)

•The position of Qposition of Q33 is

once the measurements have been ordered. If the positions are not integers, find the quartiles by interpolation.

The prices ($) of 18 brands of walking shoes:

40 60 65 65 65 68 68 70 70

70 70 70 70 74 75 75 90 95

Position of Q1 = 0.25(18 + 1) = 4.75

Position of Q3 = 0.75(18 + 1) = 14.25

Q1is 3/4 of the way between the 4th and 5th ordered measurements, or Q1 = 65 + 0.75(65 - 65) = 65.

Example

The prices ($) of 18 brands of walking shoes:

40 60 65 65 65 68 68 70 70

70 70 70 70 74 75 75 90 95

Position of Q1 = 0.25(18 + 1) = 4.75

Position of Q3 = 0.75(18 + 1) = 14.25

Q3 is 1/4 of the way between the 14th and 15th ordered measurements, or

Q3 = 74 + .25(75 - 74) = 74.25and IQR = Q3 – Q1 = 74.25 - 65 = 9.25

Example

SortedBillions Billions Ranks 33 18 1 26 18 2 24 18 3 21 18 4 19 19 5 20 20 6 18 20 7 18 20 8 52 21 9 56 22 10 27 22 11 22 23 12 18 24 13 49 26 14 22 27 15 20 32 16 23 33 17 32 49 18 20 52 19 18 56 20

First Quartile?

Median?

Third Quartile?

Range?

Interquartile Range?

Example 2

The Variance• The variancevariance is measure of variability that uses all the measurements. It measures the average deviation of the measurements about their mean.

• Flower petals:Flower petals: 5, 12, 6, 8, 14

95

45x 9

5

45x

4 6 8 10 12 14

• The variance of a populationvariance of a population of N measurements is the average of the squared deviations of the measurements about their mean .

• The variance of a samplevariance of a sample of n measurements is the sum of the squared deviations of the measurements about their mean, divided by (n – 1).

N

xi2

2 )(

N

xi2

2 )(

1

)( 22

n

xxs i

1

)( 22

n

xxs i

The Variance

The Standard Deviation

• In calculating the variance, we squared all of the deviations, and in doing so changed the scale of the measurements.

• To return this measure of variability to the original units of measure, we calculate the standard deviationstandard deviation, the positive square root of the variance.

2

2

:deviation standard Sample

:deviation standard Population

ss

2

2

:deviation standard Sample

:deviation standard Population

ss

1

)( 22

n

xxs i

5 -4 16

12 3 9

6 -3 9

8 -1 1

14 5 25

Sum 45 0 60

Use the Definition Formula:

ix ix x 2( )ix x

154

60

87.3152 ss

Two Ways to Calculate the Sample Variance

Two Ways to Calculate the Sample Variance

1

)( 22

2

nn

xx

s

ii

5 25

12 144

6 36

8 64

14 196

Sum 45 465

Use the calculation formula:

ix 2ix

154

545

4652

87.3152 ss

Some NotesSome Notes

Using Measures of Center and Spread: Chebysheff’s Theorem

Given a number k greater than or equal to 1 and a set of n measurements, at least 1-(1/k2) of the measurement will lie within k standard deviations of the mean.

Given a number k greater than or equal to 1 and a set of n measurements, at least 1-(1/k2) of the measurement will lie within k standard deviations of the mean.

Can be used for either samples ( and s) or for a population ( and ).Important results: Important results:

If k = 2, at least 1 – 1/22 = 3/4 of the measurements are within 2 standard deviations of the mean.If k = 3, at least 1 – 1/32 = 8/9 of the measurements are within 3 standard deviations of the mean.

x

26

Chebyshev’s Theorem: An exampleThe arithmetic mean biweekly amount contributed by the Dupree

Paint employees to the company’s profit-sharing plan is $51.54, and the standard deviation is $7.51. At least what percent of the contributions lie within plus 3.5 standard deviations and minus 3.5 standard deviations of the mean?

Using Measures of Center and Spread: The Empirical Rule

Given a distribution of measurements that is approximately mound-shaped:

The interval contains approximately 68% of the measurements.

The interval 2 contains approximately 95% of the measurements.

The interval 3 contains approximately 99.7% of the measurements.

Given a distribution of measurements that is approximately mound-shaped:

The interval contains approximately 68% of the measurements.

The interval 2 contains approximately 95% of the measurements.

The interval 3 contains approximately 99.7% of the measurements.

The Empirical Rule: An Example

k ks Interval Proportionin Interval

Tchebysheff Empirical Rule

1 44.9 10.73 34.17 to 55.63 31/50 (.62) At least 0 .68

2 44.9 21.46 23.44 to 66.36 49/50 (.98) At least .75 .95

3 44.9 32.19 12.71 to 77.09 50/50 (1.00)

At least .89 .997

x

•Do the actual proportions in the three intervals agree with those given by Tchebysheff’s Theorem?

•Do they agree with the Empirical Rule?

•Why or why not?

Yes. Chebysheff’s Theorem must be true for any data set.

No. Not very well.

The data distribution is not very mound-shaped, but skewed right.

The length of time for a worker to complete a specified operation averages 12.8 minutes with a standard deviation of 1.7 minutes. If the distribution of times is approximately mound-shaped, what proportion of workers will take longer than 16.2 minutes to complete the task?

.475 .475

95% between 9.4 and 16.2

47.5% between 12.8 and 16.2

.025 (50-47.5)% = 2.5% above 16.2

Example

• From Chebysheff’s Theorem and the Empirical Rule, we know that R 4-6 s

• To approximate the standard deviation of a set of measurements, we can use:

Approximating s

set.data largea for or 6/

4/

Rs

Rs

set.data largea for or 6/

4/

Rs

Rs

R = 70 – 26 = 44

114/444/ Rs

Actual s = 10.73

The ages of 50 tenured faculty at a state university.

• 34 48 70 63 52 52 35 50 37 43 53 43 52 44

• 42 31 36 48 43 26 58 62 49 34 48 53 39 45

• 34 59 34 66 40 59 36 41 35 36 62 34 38 28

• 43 50 30 43 32 44 58 53

Approximating s

3. Measures of Relative Standing

• Where does one particular measurement stand in relation to the other measurements in the data set?

• How many standard deviations away from the mean does the measurement lie? This is measured by the zz-score.-score.

s

xxz

score-

s

xxz

score-

5x 9x

s s

4

Suppose s = 2. s

x = 9 lies z =2 std dev from the mean.

Not unusualOutlier Outlier

z-Scores• From Chebysheff’s Theorem and the Empirical Rule

• At least 3/4 and more likely 95% of measurements lie within 2 standard deviations of the mean.

• At least 8/9 and more likely 99.7% of measurements lie within 3 standard deviations of the mean.

• z-scores between –2 and 2 are not unusual. z-scores should not be more than 3 in absolute value. z-scores larger than 3 in absolute value would indicate a possible outlier.outlier.

z

-3 -2 -1 0 1 2 3

Somewhat unusual

• How many measurements lie below the measurement of interest? This is measured by the ppth th percentile.percentile.

p-th percentile

(100-p) %x

p %

3. Measures of Relative Standing (cont’d)

Example

• 90% of all men (16 and older) earn more than RM 1000 per week.

BUREAU OF LABOR STATISTICS

RM 1000

90%10%

50th Percentile

25th Percentile

75th Percentile

Median

Lower Quartile (Q1)

Upper Quartile (Q3)

RM 1000 is the 10th percentile.RM 1000 is the 10th percentile.

The Five-Number Summary:

Min Q1 Median Q3 Max

The Five-Number Summary:

Min Q1 Median Q3 Max

• Divides the data into 4 sets containing an equal number of measurements.

• A quick summary of the data distribution.

• Use to form a box plotbox plot to describe the shapeshape of the distribution and to detect outliersoutliers.

Calculate Q1, the median, Q3 and IQR.

Draw a horizontal line to represent the scale of measurement.

Draw a box using Q1, the median, Q3.

QQ11 mm QQ33

QQ11 mm QQ33

Isolate outliers by calculatingLower fence: Q1-1.5 IQRUpper fence: Q3+1.5 IQR

Measurements beyond the upper or lower fence is are outliers and are marked (*).

*

LF UF

QQ11 mm QQ33

*

Draw “whiskers” connecting the largest and smallest measurements that are NOT outliers to the box.

LF UF

Amt of sodium in 8 brands of cheese:

260 290 300 320 330 340 340 520

m = 325m = 325 QQ33 = 340 = 340

mm

QQ11 = 292.5 = 292.5

QQ33QQ11

IQR = 340-292.5 = 47.5

Lower fence = 292.5-1.5(47.5) = 221.25

Upper fence = 340 + 1.5(47.5) = 411.25

mm

QQ33QQ11

*

Outlier: x = 520

LF UF

Median line in center of box and whiskers of equal length—symmetric distribution

Median line left of center and long right whisker—skewed right

Median line right of center and long left whisker—skewed left

Grouped and Ungrouped Data

Sample MeanSample Mean

Ungrouped data:

Grouped data:

nxxxx n

n

xn

ii

211

k

iii

k

iii

xfnn

xfx

1

*1

*

1

n= number of observation

n = sum of the frequencies

k= number class

fi = frequency of each class

= midpoint of each class*ix

Example: - ungrouped data

Resistance of 5 coils:

3.35, 3.37, 3.28, 3.34, 3.30 ohm.The average:

33.3

5

30.334.328.337.335.31

n

xx

n

ii

Example: - Example: - grouped datagrouped dataFrequency Distributions of the life of 320 tires in 1000 km

Boundaries Midpoint Frequency, fi

23.5-26.5 25.0 4 10026.5-29.5 28.0 36 100829.5-32.5 31.0 51 158132.5-35.5 34.0 63 214235.5-38.5 37.0 58 214638.5-41.5 40.0 52 208041.5-44.5 43.0 34 146244.5-47.5 46.0 16 73647.5-50.5 49.0 6 294

Total n = 320

n

xfx

k

iii

1

*

*ix *

ii xf

115491

*

k

iii xf

1.36320

549,11

Sample Variance

Ungrouped data:

11

)(

2

11

2

1

2

2

n

nxx

n

xxs

n

ii

n

ii

n

ii

Grouped data:

)1(

)(1

2

1

*2*

2

n

nxfxf

s

k

i

k

iiiii

Example- ungrouped data

Sample: Moisture content of kraft paper are:

6.7, 6.0, 6.4, 6.4, 5.9, and 5.8 %.

Sample standard deviation, s = 0.35 %

35.0)16(

6)2.37()26.231( 2

s

Calculating the Sample Standard Calculating the Sample Standard Deviation - Deviation - Grouped DataGrouped Data

Standard deviation for a grouped sample:

)1(

)(1

2

1

*2*

n

nxfxf

s

k

i

k

iiiii

Boundaries

72.5-81.5 77.0 5 385 29645

81.5-90.5 86.0 19 1634 140524

90.5-99.5 95 31 2945 279775

99.5-108.5 104.0 27 2808 292032

108.5-117.5 113 14 1582 178766

Total 96 9354 920742 9.9)196(

96)9354()742,920( 2

s

Table: Car speeds in km/h

*ii xf 2*

ii xf*ix if

Mode•Mode is the value that has the highest frequency in a data set.•For grouped data, class mode (or, modal class) is the class with the highest frequency.•To find mode for grouped data, use the following formula:

Mode – Grouped DataMode – Grouped Data

Where:

L = Lower boundary of the modal class

ldd

dL

21

1Mode

1d frequency of modal class –frequency of the class before=

2d frequency of modal class –frequency of the class after=l width of the modal class=

Example of Grouped Data (Mode)

Based on the grouped data below, find the mode

Class Limit Boundaries fCumFreq

Cum Rel Freq

99 - 108 98.5 - 108.5 6 6 0.150

109 - 118 108.5 - 118.5 7 13 0.325

119 - 128 118.5 - 128.5 13 26 0.650

129 - 138 128.5 - 138.5 8 34 0.850

139 - 148 138.5 - 148.5 6 40 0.975

Modal Class = 119 – 128 (third)

L = 118.5

l = 10

67131 d58132 d 10

11

65.118

Mode21

1

l

dd

dL

95.123

Percentile (Ungrouped Data)Percentile (Ungrouped Data)

•that value in a sorted list of the data that has approx p% of the measurements below it and approx (1-p)% above it. •The pth percentile from the table of frequency can be calculated as follows:

pth percentile = (p/100)*n (if i is NOT an integer, round up)

Percentile (Grouped Data)Percentile (Grouped Data)

lf

CFBkLBCpp

p

S

100

pnk

n = number of observationsLBCp = Lower Boundary of the percentile classCFB = Cumulative frequency of the class before Cpl = Class widthfp = Frequency of percentile class

Example- Percentile for grouped data

Using the previous table of freq, find the median and 80th percentile?

Class Limit Boundaries fCumFreq

Cum Rel Freq

99 - 108 98.5 - 108.5 6 6 0.150109 - 118 108.5 - 118.5 7 13 0.325119 - 128 118.5 - 128.5 13 26 0.650129 - 138 128.5 - 138.5 8 34 0.850139 - 148 138.5 - 148.5 6 40 0.975

Median, 50S

20100

5040

k

50C (119 – 128) (3rd class)

5.118pLBC

10l13CFB

13pf

1013

13205.11850S

885.128

80th percentile, 80S

32100

8040

k

80C (129 – 138) (4rd class)

5.128pLBC

10l26CFB

8pf

108

26325.12880S

136