Introduction To Statistics

Lecture 09

Dr. MUMTAZ AHMEDMTH 161: Introduction To StatisticsReview of Previous LectureIn last lecture we discussed:Measures of Central TendencyWeighted MeanCombined MeanMerits and demerits of Arithmetic MeanMedianMedian for Ungrouped Data22Objectives of Current LectureMeasures of Central TendencyMedianMedian for grouped DataMerits and demerits of MedianModeMode for Grouped DataMode for Ungrouped DataMerits and demerits of Mode33Objectives of Current LectureMeasures of Central TendencyGeometric MeanGeometric Mean for Grouped DataGeometric Mean for Ungrouped DataMerits and demerits of Geometric Mean44Median for Grouped DataMedian for Grouped DataExample: Calculate Median for the distribution of examination marks provided below:

MarksNo of Students (f)30-39840-498750-5919060-6930470-7921180-898590-9920Median for Grouped DataCalculate Class BoundariesMarksClass BoundariesNo of Students (f)30-39840-498750-5919060-6930470-7921180-898590-9920Median for Grouped DataCalculate Class BoundariesMarksClass BoundariesNo of Students (f)30-3929.5-39.5840-498750-5919060-6930470-7921180-898590-9920Median for Grouped DataCalculate Class BoundariesMarksClass BoundariesNo of Students (f)30-3929.5-39.5840-4939.5-49.58750-5949.5-59.519060-6959.5-69.630470-7969.5-79.521180-8979.5-89.58590-9989.5-99.520Median for Grouped DataCalculate Cumulative Frequency (cf)MarksClass BoundariesNo of Students (f)Cumulative Freq (cf)30-3929.5-39.58840-4939.5-49.58750-5949.5-59.519060-6959.5-69.630470-7969.5-79.521180-8979.5-89.58590-9989.5-99.520Median for Grouped DataCalculate Cumulative Frequency (cf)MarksClass BoundariesNo of Students (f)Cumulative Freq (cf)30-3929.5-39.58840-4939.5-49.5878+87=9550-5949.5-59.519060-6959.5-69.630470-7969.5-79.521180-8979.5-89.58590-9989.5-99.520Median for Grouped DataCalculate Cumulative Frequency (cf)MarksClass BoundariesNo of Students (f)Cumulative Freq (cf)30-3929.5-39.58840-4939.5-49.5879550-5949.5-59.519028560-6959.5-69.630458970-7969.5-79.521180080-8979.5-89.58588590-9989.5-99.520905Median for Grouped DataFind Median Class:Median=Marks obtained by (n/2)th student=905/2=452.5th studentLocate 452.5 in the Cumulative Freq. column.MarksClass BoundariesNo of Students (f)Cumulative Freq (cf)30-3929.5-39.58840-4939.5-49.5879550-5949.5-59.519028560-6959.5-69.630458970-7969.5-79.521180080-8979.5-89.58588590-9989.5-99.520905TotalMedian for Grouped DataFind Median Class:452.5 in the Cumulative Freq. column.Hence59.5-69.5 is the Median Class.MarksClass BoundariesNo of Students (f)Cumulative Freq (cf)30-3929.5-39.58840-4939.5-49.5879550-5949.5-59.519028560-6959.5-69.630458970-7969.5-79.521180080-8979.5-89.58588590-9989.5-99.520905Median for Grouped DataMarksClass BoundariesNo of Students (f)Cumulative Freq (cf)30-3929.5-39.58840-4939.5-49.5879550-5949.5-59.5190285=C60-69l=59.5-69.5304=f58970-7969.5-79.521180080-8979.5-89.58588590-9989.5-99.520905Merits of MedianMerits of Median are:Easy to calculate and understand.Median works well in case of Symmetric as well as in skewed distributions as opposed to Mean which works well only in case of Symmetric Distributions.It is NOT affected by extreme values.Example:Median of 1, 2, 3, 4, 5 is 3.If we change last number 5 to 20 then Median will still be 3.Hence Median is not affected by extreme values.De-Merits of MedianDe-Merits of Median are:It requires the data to be arranged in some order which can be time consuming and tedious, though now-a-days we can sort the data via computer very easily.ModeMode is a value which occurs most frequently in a data.

Mode is a French word meaning fashion, adopted for most frequent value.

Calculation:The mode is the value in a dataset which occurs most often or maximum number of times.Mode for Ungrouped DataExample 1:Marks: 10, 5, 3, 6, 10Mode=10

Example 2:Runs: 5, 2, 3, 6, 2, 11, 7Mode=2

Often, there is no mode or there are several modes in a set of data.Example: marks: 10, 5, 3, 6, 7No Mode

Sometimes we may have several modes in a set of data.Example: marks: 10, 5, 3, 6, 10, 5, 4, 2, 1, 9 Two modes (5 and 10)Mode for Qualitative DataMode is mostly used for qualitative data.

Mode is PTI

Mode for Grouped DataMode for Grouped DataExample: Calculate Mode for the distribution of examination marks provided below:

MarksNo of Students (f)30-39840-498750-5919060-6930470-7921180-898590-9920Mode for Grouped DataCalculate Class BoundariesMarksClass BoundariesNo of Students (f)30-39840-498750-5919060-6930470-7921180-898590-9920Mode for Grouped DataCalculate Class BoundariesMarksClass BoundariesNo of Students (f)30-3929.5-39.5840-498750-5919060-6930470-7921180-898590-9920Mode for Grouped DataCalculate Class BoundariesMarksClass BoundariesNo of Students (f)30-3929.5-39.5840-4939.5-49.58750-5949.5-59.519060-6959.5-69.630470-7969.5-79.521180-8979.5-89.58590-9989.5-99.520Mode for Grouped DataFind Modal Class (class with the highest frequency)MarksClass BoundariesNo of Students (f)30-3929.5-39.5840-4939.5-49.58750-5949.5-59.519060-6959.5-69.530470-7969.5-79.521180-8979.5-89.58590-9989.5-99.520Mode for Grouped DataFind Modal Class (class with the highest frequency)MarksClass BoundariesNo of Students (f)30-3929.5-39.5840-4939.5-49.58750-5949.5-59.519060-6959.5-69.530470-7969.5-79.521180-8979.5-89.58590-9989.5-99.520Mode for Grouped DataMarksClass BoundariesNo of Students (f)30-3929.5-39.5840-4939.5-49.58750-5949.5-59.5190=f160-69304=fm70-7969.5-79.5211=f280-8979.5-89.58590-9989.5-99.520Merits of ModeMerits of Mode are:Easy to calculate and understand. In many cases, it is extremely easy to locate it.It works well even in case of extreme values.It can be determined for qualitative as well as quantitative data.

De-Merits of ModeDe-Merits of Mode are:It is not based on all observations.When the data contains small number of observations, the mode may not exist.Geometric MeanWhen you want to measure the rate of change of a variable over time, you need to use the geometric mean instead of the arithmetic mean.

Calculation:The geometric mean is the nth root of the product of n values.Geometric Mean for Ungrouped DataGeometric Mean for Ungrouped DataGeometric Mean for Ungrouped DataGeometric Mean for Ungrouped DataGeometric Mean for Ungrouped DataExamples of Ungrouped Data:

Example 1: Marks obtained by 5 students, 2, 8, 4 (Alternative Method)

Geometric Mean for Ungrouped DataExamples of Ungrouped Data:

Example 1: Marks obtained by 5 students, 2, 8, 4 (Alternative Method)

Marks (x)Log(x)2Log(2)=0.3010384Geometric Mean for Ungrouped DataExamples of Ungrouped Data:

Example 1: Marks obtained by 5 students, 2, 8, 4 (Alternative Method)

Marks (x)Log(x)2Log(2)=0.3010380.9030940.60206Geometric Mean for Ungrouped DataExamples of Ungrouped Data:

Example 1: Marks obtained by 5 students, 2, 8, 4 (Alternative Method)

Geometric Mean for Ungrouped DataExamples of Ungrouped Data:

Example 1: Marks obtained by 5 students, 2, 8, 4

Geometric Mean for Ungrouped DataExamples of Ungrouped Data:

Example 1: Marks obtained by 5 students, 2, 8, 4

Geometric Mean for Ungrouped DataExamples of Ungrouped Data:

Example 1: Marks obtained by 5 students, 2, 8, 4

ReviewLets review the main concepts:Measures of Central TendencyMedianMedian for grouped DataMerits and demerits of MedianModeMode for Grouped DataMode for Ungrouped DataMerits and demerits of Mode43ReviewLets review the main concepts:Measures of Central TendencyGeometric MeanGeometric Mean for Ungrouped Data44Next LectureIn next lecture, we will study:Geometric MeanGeometric Mean for Grouped DataMerits and demerits of Geometric Mean

Harmonic MeanHarmonic Mean for Grouped DataHarmonic Mean for Ungrouped DataMerits and demerits of Harmonic Mean45