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