Frequency Distribution Frequency Distribution • Convert raw data into a data array. Convert raw data into a data array. • Construct: Construct: – a frequency distribution. a frequency distribution. – a relative frequency distribution. a relative frequency distribution. – a cumulative relative frequency a cumulative relative frequency distribution. distribution. • Construct different types of diagrams. Construct different types of diagrams. • Visually represent data by using graphs Visually represent data by using graphs and charts. and charts.
Transcript
Frequency DistributionFrequency Distribution
• Convert raw data into a data array.Convert raw data into a data array.
• Construct:Construct:– a frequency distribution.a frequency distribution.– a relative frequency distribution.a relative frequency distribution.– a cumulative relative frequency distribution.a cumulative relative frequency distribution.
• Construct different types of diagrams.Construct different types of diagrams.
• Visually represent data by using graphs Visually represent data by using graphs and charts.and charts.
• Data arrayData array– An orderly presentation of data in either An orderly presentation of data in either
ascending or descending numerical ascending or descending numerical order.order.
• Frequency DistributionFrequency Distribution– A table that represents the data in A table that represents the data in
classes and that shows the number of classes and that shows the number of observations in each class.observations in each class.
• Frequency DistributionFrequency Distribution– ClassClass - The category - The category– FrequencyFrequency - Number in each class - Number in each class– Cum frequency: Number UP To the classCum frequency: Number UP To the class– Class limits Class limits - Boundaries for each class- Boundaries for each class– Class interval Class interval - Width of each class- Width of each class– Class mark Class mark - Midpoint of each class- Midpoint of each class
• As such NO RIGID RULEAs such NO RIGID RULE• Classes normally are : 5 to 15 depending Classes normally are : 5 to 15 depending
on the dataon the data• How to set the approximate number of How to set the approximate number of
classes to classes to beginbegin constructing a frequency constructing a frequency distribution.distribution.
• K= 1+ 3.322 log NK= 1+ 3.322 log N• Class Interval= Range/ KClass Interval= Range/ K
where K = approximate number of classes to use andwhere K = approximate number of classes to use andNN = the number of observations in the data set . = the number of observations in the data set .
PrecautionsPrecautions
• No uneven classesNo uneven classes
• Avoid odd upper limitsAvoid odd upper limits
• Avoid odd intervalAvoid odd interval
• All values should have a unique classAll values should have a unique class
TestTest
• Which type of data is better?Which type of data is better?– Grouped or ungroupedGrouped or ungrouped– WHY ?WHY ?
How to Construct aHow to Construct aFrequency DistributionFrequency Distribution
. Number of classes. Number of classes Choose an approximate number of classes for Choose an approximate number of classes for
your data. Sturges’ rule can help.your data. Sturges’ rule can help.2. Estimate the class interval2. Estimate the class interval Divide the approximate number of classes Divide the approximate number of classes (from (from
Step 1) Step 1) into the range of your data to find the into the range of your data to find the approximate class interval, where the range is approximate class interval, where the range is defined as the largest data value minus the defined as the largest data value minus the smallest data value.smallest data value.
3. Determine the class interval3. Determine the class intervalRound the estimate Round the estimate (from Step 2) (from Step 2) to a convenient to a convenient value.value.
Lower Class LimitLower Class LimitDetermine the lower class limit for the first class Determine the lower class limit for the first class by selecting a convenient number that is smaller by selecting a convenient number that is smaller than the lowest data value.than the lowest data value.
5. Class Limits5. Class LimitsDetermine the other class limits by repeatedly Determine the other class limits by repeatedly adding the class width adding the class width (from Step 2) (from Step 2) to the prior to the prior class limit, starting with the lower class limit class limit, starting with the lower class limit (from Step 3)(from Step 3)..
6. Define the classes6. Define the classesUse the sequence of class limits to define the Use the sequence of class limits to define the classesclasses
Converting to a Converting to a Relative Relative Frequency DistributionFrequency Distribution
Retain the same classes defined in the Retain the same classes defined in the frequency distribution.frequency distribution.
2. Sum the total number of observations 2. Sum the total number of observations across all classes of the frequency across all classes of the frequency distribution.distribution.
3. Divide the frequency for each class by 3. Divide the frequency for each class by the total number of observations, the total number of observations, forming the percentage of data values in forming the percentage of data values in each classeach class
Example: Problem Example: Problem
• The average daily cost to community The average daily cost to community hospitals for patient stays during 1993 for hospitals for patient stays during 1993 for each of the 50 U.S. states was given in the each of the 50 U.S. states was given in the next table.next table.– a) Arrange these into a data array.a) Arrange these into a data array.
– *) Approximately how many classes would be *) Approximately how many classes would be appropriate for these data? appropriate for these data?
– c & d) Construct a frequency distribution. State c & d) Construct a frequency distribution. State interval width and class mark.interval width and class mark.
– e) Construct a histogram, a relative frequency e) Construct a histogram, a relative frequency distribution, and a cumulative relative frequency distribution, and a cumulative relative frequency distribution.distribution.
DataDataAL $775AL $775 HI 823HI 823 MA 1,036MA 1,036 NM 1,046 SD NM 1,046 SD
506506AK 1,136AK 1,136 ID 659ID 659 MI 902MI 902 NY 784NY 784 TN 859TN 859AZ 1,091AZ 1,091 IL 917IL 917 MN 652MN 652 NC 763NC 763 TX TX
1,0101,010AR 678AR 678IN 898IN 898 MS 555MS 555 ND 507ND 507 UT 1,081UT 1,081CA 1,221CA 1,221 IA 612IA 612 MO 863MO 863 OH 940OH 940 VT 676VT 676CO 961CO 961 KS 666 KS 666 MT 482MT 482 OK 797OK 797 VA 830VA 830CT 1,058CT 1,058 KY 703KY 703 NE 626NE 626 OR 1,052 WA OR 1,052 WA
1,1431,143DE 1,024DE 1,024 LA 875LA 875 NV 900NV 900 PA 861 PA 861 WV 701WV 701FL 960FL 960 ME 738 NH 976ME 738 NH 976 RI 885 WI 744RI 885 WI 744GA 775GA 775 MD 889MD 889 NJ 829NJ 829 SC 838 WY SC 838 WY
537537
•Step 1. Number of classesStep 1. Number of classes– Sturges’ Rule: approximately 7 classes.Sturges’ Rule: approximately 7 classes.
The range is: $1,221 – $482 = $739The range is: $1,221 – $482 = $739
$739/7 = $106 and $739/8 = $92$739/7 = $106 and $739/8 = $92
• Steps 2 & 3. The Class Steps 2 & 3. The Class IntervalInterval – So, if we use 8 classes, we can make each So, if we use 8 classes, we can make each
class $100 wide.class $100 wide.
• Step 4. The Lower Class LimitStep 4. The Lower Class Limit– If we start at $450, we can cover the range in 8 If we start at $450, we can cover the range in 8
classes, each class $100 in width.classes, each class $100 in width.
The first class : $450 up to $550The first class : $450 up to $550
• Steps 5 & 6. Setting Class LimitsSteps 5 & 6. Setting Class Limits$450 up to $550$450 up to $550 $850 up to $950 $850 up to $950
$550 up to $650$550 up to $650 $950 up to $1,050 $950 up to $1,050
$650 up to $750 $1,050 up to $1,150$650 up to $750 $1,050 up to $1,150
$750 up to $850 $1,150 up to $1,250$750 up to $850 $1,150 up to $1,250
Average daily costAverage daily cost NumberNumberMarkMark
$450 – under $550$450 – under $550 44 $500$500 $550 – under $650 $550 – under $650 33 $600$600 $650 – under $750 $650 – under $750 99 $700$700 $750 – under $850 $750 – under $850 99 $800$800 $850 – under $950 $850 – under $950 11 11 $900$900 $950 – under $1,050 $950 – under $1,050 7 7 $1,000 $1,000 $1,050 – under $1,150 $1,050 – under $1,150 6 6 $1,100 $1,100 $1,150 – under $1,250 $1,150 – under $1,250 1 1 $1,200 $1,200 Interval width: $100Interval width: $100
Measures of Central Measures of Central Tendency and DispersionTendency and Dispersion
IntroductionIntroduction
Raw Data are the raw materials that will Raw Data are the raw materials that will have to be converted into finished products have to be converted into finished products (Information). From a voluminous database (Information). From a voluminous database containing raw data, it is impossible to see containing raw data, it is impossible to see any pattern unless they are converted into any pattern unless they are converted into information by data reduction. The reduction information by data reduction. The reduction can be achieved by summary measures, can be achieved by summary measures, which are concise and yet give a reasonably which are concise and yet give a reasonably accurate view of the original data. Here we accurate view of the original data. Here we cover the important summary measures of cover the important summary measures of central tendencycentral tendency and and dispersiondispersion (variation) (variation)
OutlineOutline1)1) What is Central Tendency? What is Central Tendency?
2)2) Measures of Central TendencyMeasures of Central Tendency
3)3) Measures of DispersionMeasures of Dispersion
1) What is Central 1) What is Central Tendency?Tendency?
Whenever you measure things of the same Whenever you measure things of the same kind, a fairly large number of such kind, a fairly large number of such measurements will tend to cluster around measurements will tend to cluster around the middle value. The question that arises is the middle value. The question that arises is " is it possible to define one typical " is it possible to define one typical representative average in such a manner representative average in such a manner that the remaining items in the data set will that the remaining items in the data set will cluster around this value?" will have a cluster around this value?" will have a tendency to be closer to this value? Such a tendency to be closer to this value? Such a value is called a measure of "Central value is called a measure of "Central Tendency". The other terms that are used Tendency". The other terms that are used synonymously are "Measures of Location", or synonymously are "Measures of Location", or "Statistical Averages". "Statistical Averages".
2) Measures of Central 2) Measures of Central TendencyTendency
Quantitative Specialists, Statisticians, and Quantitative Specialists, Statisticians, and Information Analysts rely heavily on summary Information Analysts rely heavily on summary measures when a large mass of data will have to measures when a large mass of data will have to be analyzed to help decision-makers. As a be analyzed to help decision-makers. As a manager, You need these summary measures of manager, You need these summary measures of central tendency to draw meaningful conclusions in central tendency to draw meaningful conclusions in your functional area of operation. The most widely your functional area of operation. The most widely used measures of central tendency are used measures of central tendency are Arithmetic Arithmetic MeanMean, , MedianMedian, and , and ModeMode..
Arithmetic MeanArithmetic MeanArithmetic Mean (called mean) is the most common Arithmetic Mean (called mean) is the most common measure of central tendency used by all managers in measure of central tendency used by all managers in their sphere of activities. It is defined as the sum of their sphere of activities. It is defined as the sum of all observations in a data set divided by the total all observations in a data set divided by the total number of observations. For example, consider a number of observations. For example, consider a data set containing the following observations:data set containing the following observations:
4, 3, 6, 5, 3, 3. The arithmetic mean = 4, 3, 6, 5, 3, 3. The arithmetic mean = (4+3+6+5+3+3)/6 =4. In symbolic form mean is (4+3+6+5+3+3)/6 =4. In symbolic form mean is given bygiven by
= Arithmetic Mean= Arithmetic Mean
= Indicates sum all X values in the data set = Indicates sum all X values in the data set
= Total number of observations(Sample Size) = Total number of observations(Sample Size)
n
XX
X
X
n
Arithmetic Mean for Raw Arithmetic Mean for Raw Data ExampleData Example
The inner diameter of a particular grade of tire The inner diameter of a particular grade of tire based on 5 sample measurements are as follows: based on 5 sample measurements are as follows: (figures in millimeters)(figures in millimeters) 565, 570, 572, 568, 585565, 570, 572, 568, 585 Applying the formulaApplying the formula
We get mean = (565+570+572+568+585)/5 =572We get mean = (565+570+572+568+585)/5 =572
Caution: Arithmetic Mean is affected by extreme Caution: Arithmetic Mean is affected by extreme values or fluctuations in sampling. It is not the best values or fluctuations in sampling. It is not the best average to use when the data set contains extreme average to use when the data set contains extreme values (Very high or very low values).values (Very high or very low values).
n
XX
MedianMedianMedian is the middle most observation when you arrange Median is the middle most observation when you arrange data in ascending or descending order of magnitude. That data in ascending or descending order of magnitude. That is, the data are ranked and the middle value is picked up. is, the data are ranked and the middle value is picked up. Median is such that 50% of the observations are above the Median is such that 50% of the observations are above the median and 50% of the observations are below the median and 50% of the observations are below the median. median. Median is a very useful measure for ranked data in the Median is a very useful measure for ranked data in the context of consumer preferences and rating. It is not context of consumer preferences and rating. It is not affected by extreme values but affected by the number of affected by extreme values but affected by the number of observations.observations.
th value of ranked datath value of ranked data
n = Number of observations in the samplen = Number of observations in the sample
Note: If the sample size is an odd number then median is Note: If the sample size is an odd number then median is (n+1)/2 th value in the ranked data. If the sample size is (n+1)/2 th value in the ranked data. If the sample size is even, then median will be between two middle values. You even, then median will be between two middle values. You take the average of these two middle values.take the average of these two middle values.
2
1nMedian
Median for Raw Data Median for Raw Data Example -Odd Sample Example -Odd Sample Size Size
Marks obtained by 7 students in Computer Science Marks obtained by 7 students in Computer Science ExamExam
are given below: Compute the median.are given below: Compute the median. 4545 4040 6060 8080 9090 6565 5555 Arranging the data after ranking givesArranging the data after ranking gives 9090 8080 6565 6060 5555 4545 4040 Median = (n+1)/2 th value in this set = (7+1)/2 thMedian = (n+1)/2 th value in this set = (7+1)/2 thobservation= 4observation= 4th th observation=60observation=60Hence Median = 60 for this problem.Hence Median = 60 for this problem.
Median for Raw Data Median for Raw Data Example - Even Sample Example - Even Sample SizeSizeDiameter of a shaft in millimeters in a manufacturing unit isDiameter of a shaft in millimeters in a manufacturing unit is
Given below for 10 samples. Calculate the median value.Given below for 10 samples. Calculate the median value.
The median falls between 5th and 6th observation. That isThe median falls between 5th and 6th observation. That is
between 2.55 and 2.56. Hence median = (2.55+2.56)/2 between 2.55 and 2.56. Hence median = (2.55+2.56)/2 =2.555=2.555
ModModee
Mode is that value which occurs most often. It has the Mode is that value which occurs most often. It has the maximum frequency of occurrence. Mode is not affected maximum frequency of occurrence. Mode is not affected by extreme values. by extreme values.
Mode is a very useful measure when you want to keep in Mode is a very useful measure when you want to keep in the inventory, the most popular shirt in terms of collar the inventory, the most popular shirt in terms of collar size during festival season. Median and mean will not be size during festival season. Median and mean will not be helpful in this type of situation. Another example where helpful in this type of situation. Another example where mode is the only answer is in determining the most mode is the only answer is in determining the most typical shoe size to be kept in stock in a shop selling typical shoe size to be kept in stock in a shop selling shoes. shoes.
Caution: In a few problems in real life, there will be more Caution: In a few problems in real life, there will be more than one mode such as bimodal and multi-modal values. than one mode such as bimodal and multi-modal values. In these cases mode cannot be uniquely determined.In these cases mode cannot be uniquely determined.
Mode for Raw Data Mode for Raw Data ExampleExample
The life in number of hours of 10 flashlight batteries The life in number of hours of 10 flashlight batteries are as follows: Find the mode.are as follows: Find the mode.
340 occurs five times. Hence, mode=340.340 occurs five times. Hence, mode=340.
Mean for Grouped DataMean for Grouped Data
Formula for Mean is given byFormula for Mean is given by
WhereWhere = Mean= Mean
= Sum of cross products of frequency in each = Sum of cross products of frequency in each class class with midpoint X of each class with midpoint X of each class
n n = Total number of observations (Total = Total number of observations (Total frequency) =frequency) =
n
fXX
X
fX
f
Mean for Grouped DataMean for Grouped DataExampleExample
Find the arithmetic mean for the Find the arithmetic mean for the following continuousfollowing continuous
frequency distribution:frequency distribution: Class Class 0-1 0-1 1-2 1-2 2-3 2-3 3-4 3-4
Applying the formulaApplying the formula==75.5/25=3.0275.5/25=3.02
A B C D 1 Class X f fX 2 0-1 0.5 1 0.5 3 1-2 1.5 4 6.0 4 2-3 2.5 8 20.0 5 3-4 3.5 7 24.5 6 4-5 4.5 3 13.5 7 5-6 5.5 2 11.0 8 Totals 25 75.5 9 Mean 3.02
n
fXX
Mean by short cut methodMean by short cut method
• Where A is Assumed value ( one can Where A is Assumed value ( one can assume any value)assume any value)
• d is the deviation of each mid-value from A. d is the deviation of each mid-value from A. If d= ( X—A)/ c , then in the formula the If d= ( X—A)/ c , then in the formula the second term is multiplied by c. Where c is second term is multiplied by c. Where c is the class interval.the class interval.
n
fdX A
Assignment: find the mean using short-cut methodAssignment: find the mean using short-cut method
Example of short cut Example of short cut methodmethod• Table here Table here
presents the profit presents the profit of 1400 companies of 1400 companies .Find the mean .Find the mean using two different using two different methodsmethods
• Sum of deviations from mean is Sum of deviations from mean is always zero.always zero.
• Sum of squared deviation from Mean Sum of squared deviation from Mean is Minimumis Minimum
• If X= X1 + X2, Then the Mean of X is If X= X1 + X2, Then the Mean of X is equal to the sum of means of X1 and equal to the sum of means of X1 and X2 (If the observations are equal)X2 (If the observations are equal)
• From two or more groups a “pooled” From two or more groups a “pooled” mean can be calculated mean can be calculated
Median for Median for Grouped Data Grouped Data
Formula for Median is given byFormula for Median is given by
Median =Median =
Where Where
L =Lower limit of the median classL =Lower limit of the median class
n = Total number of observations = n = Total number of observations =
m= Cumulative frequency preceding the median m= Cumulative frequency preceding the median classclass
f= Frequency of the median classf= Frequency of the median class
c= Class interval of the median classc= Class interval of the median class
cf
m(n/2)L
f
Median for Grouped Median for Grouped Data ExampleData Example
Find the median for the following continuousFind the median for the following continuous
• Find the Median and Mode for the following Find the Median and Mode for the following data ( salary structure of 1500 employees)data ( salary structure of 1500 employees)
Comparison of Comparison of Mean, Median, Mean, Median, ModeMode
MeanMean MedianMedian ModeModeDefined as the Defined as the arithmetic average of arithmetic average of all observations in the all observations in the data set.data set.
Requires measurement Requires measurement on all observations.on all observations.
Uniquely and Uniquely and comprehensively comprehensively defined.defined.
Defined as the Defined as the middle value in middle value in the data set the data set arranged in arranged in ascending or ascending or descending descending order.order.
Does not require Does not require measurement on measurement on all observationsall observations
Cannot be Cannot be determined determined under all under all conditions.conditions.
Defined as the most Defined as the most frequently occurring frequently occurring value in the value in the distribution; it has distribution; it has the largest the largest frequency.frequency.
Does not require Does not require measurement on all measurement on all observationsobservations
Not uniquely defined Not uniquely defined for multi-modal for multi-modal situations. situations.
Comparison of Comparison of Mean, Median, Mode Mean, Median, Mode Cont.Cont.
MeanMean MedianMedian ModeModeAffected by extreme Affected by extreme values. values.
Can be treated Can be treated algebraically. That is, algebraically. That is, Means of several Means of several groups can be groups can be combined.combined.
Not affected by Not affected by extreme values.extreme values.
Cannot be treated Cannot be treated algebraically. That algebraically. That is, Medians of is, Medians of several groups several groups cannot be cannot be combinedcombined. .
Not affected by Not affected by extreme values.extreme values.
Cannot be treated Cannot be treated algebraically. That algebraically. That is, Modes of several is, Modes of several groups cannot be groups cannot be combined.combined.
Which central tendency to Which central tendency to use…use…
• Type of data: Type of data: – If data is badly skewed: Avoid the MeanIf data is badly skewed: Avoid the Mean– If gaps in the data: Avoid medianIf gaps in the data: Avoid median– If uneven frequencies: Avoid ModeIf uneven frequencies: Avoid Mode
• Purpose of Analysis:Purpose of Analysis:– Representative value: MeanRepresentative value: Mean– Qualitative/ nominal variable: ModeQualitative/ nominal variable: Mode– Partition point: MedianPartition point: Median
Which central tendency to Which central tendency to use…use…
• Frequency distribution:Frequency distribution:– Open ended classes: Median or ModeOpen ended classes: Median or Mode– (except certain situations)(except certain situations)– Others : MeanOthers : Mean
• Nature of data:Nature of data:– Time series data: Avoid MeanTime series data: Avoid Mean– Ratios/rates : Avoid Mean Ratios/rates : Avoid Mean
RelationshipRelationship
• Mean, Median and mode are related Mean, Median and mode are related as follows:as follows:
• (Mean –Mode)= 3 ( Mean – Median)(Mean –Mode)= 3 ( Mean – Median)
• For a completely symmetric For a completely symmetric distribution, distribution,
( Normal distribution) , the three ( Normal distribution) , the three measures coincide with each other.measures coincide with each other.
Fractiles / QuantilesFractiles / Quantiles
• A FRACTILE is the value of an A FRACTILE is the value of an observation which is located at a observation which is located at a specified place in a series of data. For specified place in a series of data. For example : Median, which is located in example : Median, which is located in the middle.the middle.
• Various fractiles used are : Quartiles, Various fractiles used are : Quartiles, Deciles, Percentiles.Deciles, Percentiles.
• Median is 50Median is 50thth percentile or 5 percentile or 5thth decile or decile or 22ndnd quartile. quartile.
How to calculate fractile How to calculate fractile valuesvalues• Qn= P 25 n= Qn= P 25 n= c
f
m(nN/4)L
D n= P 10 n= D n= P 10 n= cf
m(nN/10)L
Class assignment: Calculate the fractiles Class assignment: Calculate the fractiles from the datafrom the data
AgAgee
18-18-2222
22-22-2626
26-26-3030
30-30-3434
34-34-3838
38-38-4242
42-42-4646
46-46-5050
50-50-5454
54-54-5858
FreFreqq
121200
121255
282800
262600
151555
181844
161622
8686 7575 5353
3) Measures of 3) Measures of DispersionDispersion
In simple terms, measures of dispersion In simple terms, measures of dispersion indicate how large the spread of the indicate how large the spread of the distribution is around the central tendency. distribution is around the central tendency. It answers unambiguously the question " It answers unambiguously the question " What is the magnitude of departure from the What is the magnitude of departure from the average value for different groups having average value for different groups having identical averages?". It is important to study identical averages?". It is important to study the central tendency along with dispersion the central tendency along with dispersion to throw light on the shape of the curve; to to throw light on the shape of the curve; to gauge whether there is distortion to the bell gauge whether there is distortion to the bell shaped symmetrical normal distribution shaped symmetrical normal distribution curve that forms the foundation stone upon curve that forms the foundation stone upon which the entire statistical inference is built.which the entire statistical inference is built.
RangeRange
RangeRange is the simplest of all measures of dispersion. is the simplest of all measures of dispersion. It is calculated as the difference between maximum It is calculated as the difference between maximum and minimum value in the data set.and minimum value in the data set.
Range = Range =
Example for Computing RangeExample for Computing Range The following data represent the percentage return The following data represent the percentage return on investment for 10 mutual funds per annum. on investment for 10 mutual funds per annum. Calculate Range.Calculate Range. 12, 14, 11, 18, 10.5, 11.3, 12, 14, 11, 912, 14, 11, 18, 10.5, 11.3, 12, 14, 11, 9 Range = = 18-9=9Range = = 18-9=9
MinimumMaximum XX
MinimumMaximum XX
• Range is an absolute measure and is Range is an absolute measure and is defined for a particular data set. It can not defined for a particular data set. It can not be used for comparison of two data sets.be used for comparison of two data sets.
• Coefficient of Range is an absolute Coefficient of Range is an absolute measure measure
• Coefft. Of Range= ( L—S)/ ( L+ S)Coefft. Of Range= ( L—S)/ ( L+ S)• If it’s a small value, dispersion is lessIf it’s a small value, dispersion is less• Coeftt. Of Range is not a consistent Coeftt. Of Range is not a consistent
measure, thus it is not used always.measure, thus it is not used always.– Example : two samples : first with extreme Example : two samples : first with extreme
values as 1 and 2 , the second sample having values as 1 and 2 , the second sample having extreme values as extreme values as
11 and 12. these samples’ coefficient of 11 and 12. these samples’ coefficient of range will be First sample :(2-1)/(2+1)= 1/3 , range will be First sample :(2-1)/(2+1)= 1/3 , second sample:( 12-11)/ (12+11)=1/23second sample:( 12-11)/ (12+11)=1/23
RangeRange
Caution: Caution: Range is a good measure of spread in Range is a good measure of spread in the distribution only when a data set shows a the distribution only when a data set shows a stable pattern of variation without extreme stable pattern of variation without extreme values. If one of the components of range namely values. If one of the components of range namely the maximum value or minimum value becomes the maximum value or minimum value becomes an extreme value, then range should not be used. an extreme value, then range should not be used.
Interquartile Interquartile RangeRange
Range is entirely dependent on maximum Range is entirely dependent on maximum and minimum values in the data set and is and minimum values in the data set and is highly misleading when one of them is an highly misleading when one of them is an extreme value. To overcome this deficiency, extreme value. To overcome this deficiency, you can resort to interquartile range. It is you can resort to interquartile range. It is computed as the range after eliminating the computed as the range after eliminating the highest and lowest 25% of observations in a highest and lowest 25% of observations in a data set that is arranged in ascending order. data set that is arranged in ascending order. Thus this measure is not sensitive to Thus this measure is not sensitive to extreme values.extreme values. Interquartile range = Range computed on Interquartile range = Range computed on middle 50% of the observationsmiddle 50% of the observations
Interquartile Range-Interquartile Range-ExampleExampleThe following data represent the percentage The following data represent the percentage
return on investment for 9 mutual funds per return on investment for 9 mutual funds per annum. Calculate interquartile range.annum. Calculate interquartile range.
Data Set: 12, 14, 11, 18, 10.5, 12, 14, 11, 9Data Set: 12, 14, 11, 18, 10.5, 12, 14, 11, 9Arranging in ascending order, the data set Arranging in ascending order, the data set becomesbecomes
9, 10.5, 11, 11, 12, 12, 14, 14, 189, 10.5, 11, 11, 12, 12, 14, 14, 18 Ignore the first two (9, 10.5) and last two (14, Ignore the first two (9, 10.5) and last two (14, 18) observations in this data set. The remaining 18) observations in this data set. The remaining contains 50% of the data. They are 11, 11, 12, contains 50% of the data. They are 11, 11, 12, 12, 14, and 14. For this if you calculate range, 12, 14, and 14. For this if you calculate range, you get interquartile range.you get interquartile range. Interquartile range = 14-11 =3. Interquartile range = 14-11 =3.
• This is an absolute measure of This is an absolute measure of dispersion, not to be used for dispersion, not to be used for comparisoncomparison
• For comparison we use “ Coefficient For comparison we use “ Coefficient of Quartile deviation”of Quartile deviation”
• Coefft. Of QD = ( Q3—Q1)/( Q3 +Q1)Coefft. Of QD = ( Q3—Q1)/( Q3 +Q1)
Mean Absolute Mean Absolute Deviation(MAD)Deviation(MAD)
Mean Absolute Deviation (MAD) is defined as the average based on Mean Absolute Deviation (MAD) is defined as the average based on the deviations measured from arithmetic mean, in which all the deviations measured from arithmetic mean, in which all deviations are treated as positive ignoring the actual sign. Unlike deviations are treated as positive ignoring the actual sign. Unlike range, MAD is based on all observations. Hence it reflects the range, MAD is based on all observations. Hence it reflects the dispersion of every item in the distribution. In symbolic form, it is dispersion of every item in the distribution. In symbolic form, it is defined by the following formula.defined by the following formula. MAD = MAD =
WhereWhere
represents sum of all deviations from arithmetic mean represents sum of all deviations from arithmetic mean after after ignoring signignoring sign
= Arithmetic Mean = Arithmetic Mean n = Number of observations in the sample(sample size)n = Number of observations in the sample(sample size)
Caution: Mean Absolute Deviation (MAD) has two weaknesses. 1) It Caution: Mean Absolute Deviation (MAD) has two weaknesses. 1) It cannot be combined for several groups. 2) Ignoring the sign has cannot be combined for several groups. 2) Ignoring the sign has serious implications to a business manager attempting to measure serious implications to a business manager attempting to measure the spread of the distribution in a scientific manner.the spread of the distribution in a scientific manner.
n
XX
XX
X
Example for MADExample for MAD
The following data represent the percentage return on The following data represent the percentage return on investment for 10 mutual funds per annum. Calculate investment for 10 mutual funds per annum. Calculate MAD (Please note that this is the same example used for MAD (Please note that this is the same example used for computing Rangecomputing Range))
Standard deviation forms the basis for the discussion on Standard deviation forms the basis for the discussion on Inferential Statistics. It is a classic measure of dispersion. Inferential Statistics. It is a classic measure of dispersion. It has many advantages over the rest of the measures of It has many advantages over the rest of the measures of variations. It is based on all observations. It is capable of variations. It is based on all observations. It is capable of being algebraically treated which implies that you can being algebraically treated which implies that you can combine standard deviations of many groups. It plays a combine standard deviations of many groups. It plays a very vital role in testing hypotheses and forming very vital role in testing hypotheses and forming confidence interval. confidence interval.
To define standard deviation, you need to define another To define standard deviation, you need to define another term called variance. In simple terms, standard deviation term called variance. In simple terms, standard deviation is the square root of variance. is the square root of variance.
Important Terms with Important Terms with NotationsNotations
Im p o rta n t T e rm s w ith n o ta t io n s
K e y R e m a rk s
S a m p le V a r ia n c e 1
)(2
2
n
XXS
S a m p le S t a n d a rd D e v ia t io n
S =1
)(2
n
XX
P o p u la t io n V a r ia n c e
2=
N
X 2)(
P o p u la t io n S t a n d a rd D e v ia t io n
N
X 2)(
W h e re n
XX (S a m p le M e a n ) a n d
N
X (P o p u la t io n M e a n )
n = N u m b e r o f o b se rv a t io n s in t h e sa m p le (S a m p le s iz e ) N = N u m b e r o f o b se rv a t io n s in t h e P o p u la t io n (P o p u la t io n S iz e )
1 . 1
)(2
2
n
XXS is a n u n b ia se d
e s t im a to r o f 2=
N
X 2)(
2 . n
XX is a n u n b ia se d
e s t im a to r o f N
X
3 . T h e d iv iso r n -1 is a lw a ys u se d w h ile c a lc u la t in g sa m p le v a r ia n c e fo r e n su r in g p ro p e r ty o f b e in g u n b ia se d
4 . S t a n d a rd d e v ia t io n is a lw a ys th e
sq u a re ro o t o f v a r ia n c e
Example for Standard Example for Standard DeviationDeviation
The following data represent the percentage The following data represent the percentage return on investment for 10 mutual funds return on investment for 10 mutual funds per annum. Calculate the sample standard per annum. Calculate the sample standard deviation.deviation.
Solution for the Solution for the Example for the Example for the
ExampleExample
Solution for the Solution for the Example Cont.Example Cont.
From the spreadsheet of Microsoft Excel in the previous From the spreadsheet of Microsoft Excel in the previous slide, it is easy to see slide, it is easy to see
that Mean = that Mean = =12.28 (In column A and row14, =12.28 (In column A and row14, 12.28 is seen).12.28 is seen).
Sample Variance = =6.33 (In column D and Sample Variance = =6.33 (In column D and row 14, 6.33 is seen)row 14, 6.33 is seen)
Sample Standard Deviation = S = = Sample Standard Deviation = S = = 2.522.52
(In column D and row 15, 2.52 is seen)(In column D and row 15, 2.52 is seen)
n
XX
1n
)X(XS
22
1n
)X(X 2
Standard Deviation Standard Deviation for Grouped Datafor Grouped Data
The standard deviation for sample data, based on The standard deviation for sample data, based on frequency distribution is given byfrequency distribution is given by
S = which is used to estimate theS = which is used to estimate the
Population Standard Deviation .Population Standard Deviation .
Here Here
n is the Sample Size = , X =Mid Point of each n is the Sample Size = , X =Mid Point of each classclass
22
][n
d
n
d
n
fXX
f
Standard Deviation for Standard Deviation for Grouped Data-Grouped Data-ExampleExample
Frequency Distribution of Return on Investment Frequency Distribution of Return on Investment of Mutual Fundsof Mutual Funds
Return on Return on InvestmentInvestment
Number of Number of Mutual FundsMutual Funds
5-105-10
10-1510-15
15-2015-20
20-2520-25
25-3025-30
TotalTotal
1010
1212
1616
1414
88
6060
Solution for the Solution for the ExampleExample
Solution for the Solution for the ExampleExample
From the spreadsheet of Microsoft Excel in the From the spreadsheet of Microsoft Excel in the previousprevious
slide, it is easy to seeslide, it is easy to see
Mean = =1040/60=17.333(cell Mean = =1040/60=17.333(cell F10), F10),
Standard Deviation = S = = Standard Deviation = S = = = 6.44 = 6.44
(Cell H12) (Cell H12)
n
fXX
X2
n
X2
59
2448.33
Calculation of SD : Raw Calculation of SD : Raw datadata• First the direct First the direct
methodmethod
• Without deviation Without deviation methodmethod
• Assumed mean Assumed mean
methodmethod
X2
n
X2
22
][n
d
n
d
1n
)X(X 2
Calculation of SD : Grouped Calculation of SD : Grouped
• First the direct First the direct methodmethod
• Without deviation Without deviation methodmethod
• Assumed mean Assumed mean methodmethod
1n
)Xf(X 2
X2
n
fX2
22
][n
fd
n
fd
Class assignmentClass assignment
• Find the average Find the average deviation and deviation and standard deviation standard deviation of the following of the following data:data:
dispersion of measuresother and S.D between ipRelationsh
5432
nsobservatio the of % 68.26 covers S.D
nsobservatio the of 57.5% covers A.D
X
X
Coefficient of VariationCoefficient of Variation(Relative Dispersion)(Relative Dispersion)
Coefficient of Variation (CV) is defined as the ratio of Coefficient of Variation (CV) is defined as the ratio of Standard Deviation to Mean. Standard Deviation to Mean. In symbolic formIn symbolic form
CV= for the sample data and = for the CV= for the sample data and = for the population data. population data.
CV is the measure to use when you want to see the CV is the measure to use when you want to see the relative spread across groups or segments. It also relative spread across groups or segments. It also measures the extent of spread in a distribution as a measures the extent of spread in a distribution as a percentage to the mean. Larger the CV, greater is percentage to the mean. Larger the CV, greater is the percentage spread. As a manager, you would like the percentage spread. As a manager, you would like to have a small CV so that your assessment in a to have a small CV so that your assessment in a situation is robust. The percentage risk is minimized.situation is robust. The percentage risk is minimized.
X
Sμ
σ
Coefficient of Coefficient of VariationVariationExampleExample
Consider two Sales Persons working in the same Consider two Sales Persons working in the same territory. The sales performance of these two in territory. The sales performance of these two in the context of selling PCs are given below. the context of selling PCs are given below. Comment on the results. Comment on the results.
Sales Person 1 Sales Person 2 Mean Sales (One year average) 50 units Standard Deviation 5 units
Mean Sales (One year average)75 units Standard deviation 25 units
Interpretation for the Interpretation for the Example Example
The CV is 5/50 =0.10 or 10% for the Sales Person1 The CV is 5/50 =0.10 or 10% for the Sales Person1 and 25/75=0.33 or 33% for sales Person2. It and 25/75=0.33 or 33% for sales Person2. It seems Sales Person1 performs better than Sales seems Sales Person1 performs better than Sales Person2 with less relative dispersion or scattering. Person2 with less relative dispersion or scattering. Sales Person2 has a very high departure or Sales Person2 has a very high departure or standard deviation from his average sales standard deviation from his average sales achievement. The moral of the story is "don't get achievement. The moral of the story is "don't get carried away by absolute number". Look at the carried away by absolute number". Look at the scatter. Even though, Sales Person2 has achieved scatter. Even though, Sales Person2 has achieved a higher average, his performance is not a higher average, his performance is not consistent and seems erratic.consistent and seems erratic.
Example:Coefficient of VariationExample:Coefficient of Variation
• Since Mean and variance are Since Mean and variance are enough to compare two groups of enough to compare two groups of data CV is used to measure the data CV is used to measure the relative spread of the datarelative spread of the data
• Two factories which have 50 and Two factories which have 50 and 100 employees have the average 100 employees have the average wages as Rs.120 per day and Rs. wages as Rs.120 per day and Rs. 85 per day. The variance of wages 85 per day. The variance of wages in the two factories are 9 and 16 in the two factories are 9 and 16 respectively. Find which factory has respectively. Find which factory has more uniformity in wages? more uniformity in wages?
• CV for factory A = 3/120x 100= 2.5CV for factory A = 3/120x 100= 2.5
• CV for factory B= 4/85x 100= 4.7CV for factory B= 4/85x 100= 4.7
• Factory A has more uniform wagesFactory A has more uniform wages
SkewnessSkewness– Measure of asymmetry of a frequency Measure of asymmetry of a frequency
distributiondistribution•Skewed to leftSkewed to left•Symmetric or unskewedSymmetric or unskewed•Skewed to rightSkewed to right
KurtosisKurtosis– Measure of flatness or peakedness of a Measure of flatness or peakedness of a
KurtosisKurtosis• Kurtosis is measured by Beta2Kurtosis is measured by Beta2
• Beta2= (Mu4)/ (Mu2)^2Beta2= (Mu4)/ (Mu2)^2
• Where Mu2= (1/N) Sum(X-mean)^2Where Mu2= (1/N) Sum(X-mean)^2– And Mu4= (1/N) Sum (X-Mean)^4And Mu4= (1/N) Sum (X-Mean)^4
KurtosisKurtosis
• PlatyKurtic : FlatPlatyKurtic : Flat
• Mesokurtic: NormalMesokurtic: Normal
• Leptokurtic: Very highLeptokurtic: Very high
• Beta2= Mu4/(Mu2)^2Beta2= Mu4/(Mu2)^2
• Where Mu4= 1/n( Sum fd^4)Where Mu4= 1/n( Sum fd^4)
• and Mu2= 1/n( Sum fd^2)and Mu2= 1/n( Sum fd^2)
Chebyshev’s TheoremChebyshev’s TheoremApplies to Applies to any any distribution, regardless of distribution, regardless of
shapeshapePlaces lower limits on the percentages of Places lower limits on the percentages of
observations within a given number of observations within a given number of standard deviations from the meanstandard deviations from the mean
Empirical RuleEmpirical RuleApplies only to roughly Applies only to roughly mound-shapedmound-shaped and and
symmetricsymmetric distributions distributionsSpecifies approximate percentages of Specifies approximate percentages of
observations within a given number of observations within a given number of standard deviations from the mean standard deviations from the mean
Relations between the Mean Relations between the Mean and Standard Deviationand Standard Deviation
11
21
14
34
75%
11
31
19
89
89%
11
41
116
1516
94%
2
2
2
At least of the elements of At least of the elements of anyany distribution lie within distribution lie within kk standard deviations of the meanstandard deviations of the mean
At least
Lie within
Standarddeviationsof the mean
2
3
4
Chebyshev’s TheoremChebyshev’s Theorem
2
11k
For roughly mound-shaped and symmetric distributions, approximately:
68% 1 standard deviation of the mean
95% Lie within
2 standard deviations of the mean
All 3 standard deviations of the mean
Empirical RuleEmpirical Rule
Pie ChartsPie ChartsCategories represented as percentages of totalCategories represented as percentages of total
Bar GraphsBar GraphsHeights of rectangles represent group Heights of rectangles represent group
frequenciesfrequencies Frequency PolygonsFrequency Polygons
Height of line represents frequency Height of line represents frequency OgivesOgives
Height of line represents cumulative frequencyHeight of line represents cumulative frequency Time PlotsTime Plots
Represents values over timeRepresents values over time
1-8 Methods of Displaying 1-8 Methods of Displaying DataData
Pie ChartPie Chart
Bar ChartBar Chart
Average Revenues
Average Expenses
Fig. 1-11 Airline Operating Expenses and Revenues
1 2
1 0
8
6
4
2
0
A i r li n e
American Continental Delta Northwest Southwest United USAir
Relative Frequency Polygon Ogive
Frequency Polygon and Frequency Polygon and OgiveOgive
50403020100
0.3
0.2
0.1
0.0
Re
lativ
e F
req
ue
ncy
Sales50403020100
1.0
0.5
0.0
Cu
mu
lativ
e R
ela
tive
Fre
qu
en
cySales
OSAJJMAMFJDNOSAJJMAMFJDNOSAJJMAMFJ
8.5
7.5
6.5
5.5
Month
Mill
ions
of T
ons
M o nthly S te e l P ro d uc tio n
(P ro b le m 1 -4 6 )
Time PlotTime Plot
Stem-and-Leaf DisplaysStem-and-Leaf Displays Quick-and-dirty listing of all observations Quick-and-dirty listing of all observations Conveys some of the same information as a histogramConveys some of the same information as a histogram
Box PlotsBox Plots MedianMedian Lower and upper quartilesLower and upper quartiles Maximum and minimumMaximum and minimum
Techniques to determine relationships and trends, identify outliers and influential observations, and quickly describe or summarize data sets.
Techniques to determine relationships and trends, identify outliers and influential observations, and quickly describe or summarize data sets.
1-9 Exploratory Data 1-9 Exploratory Data Analysis - EDAAnalysis - EDA
Example 1-8: Stem-and-Leaf Example 1-8: Stem-and-Leaf DisplayDisplay
X X *o
MedianQ1 Q3InnerFence
InnerFence
OuterFence
OuterFence
Interquartile Range
Smallest data point not below inner fence
Largest data point not exceeding inner fence
Suspected outlierOutlier
Q1-3(IQR)Q1-1.5(IQR) Q3+1.5(IQR)
Q3+3(IQR)
Elements of a Box PlotElements of a Box Plot
Box PlotBox Plot
Example: Box Plot Example: Box Plot
1-10 Using the Computer – 1-10 Using the Computer – The Template OutputThe Template Output
Using the Computer – Using the Computer – Template Output for the Template Output for the HistogramHistogram
Using the Computer – Using the Computer – Template Output for Template Output for Histograms for Grouped Histograms for Grouped DataData
Using the Computer – Template Output for Using the Computer – Template Output for Frequency Polygons & the Ogive for Grouped Frequency Polygons & the Ogive for Grouped DataData
Using the Computer – Template Output Using the Computer – Template Output for Two Frequency Polygons for Grouped for Two Frequency Polygons for Grouped DataData
Using the Computer – Pie Using the Computer – Pie Chart Template OutputChart Template Output
Using the Computer – Bar Using the Computer – Bar Chart Template OutputChart Template Output
Using the Computer – Box Using the Computer – Box Plot Template OutputPlot Template Output
Using the Computer – Box Using the Computer – Box Plot Template to Compare Plot Template to Compare Two Data SetsTwo Data Sets
Using the Computer – Using the Computer – Time Plot Template Time Plot Template
Using the Computer – Using the Computer – Time Plot Comparison Time Plot Comparison Template Template
