Kinds or Branches Statistics

Statistics may be divided into two main branches:(1) Descriptive Statistics (2) Inferential Statistics

(1) Descriptive Statistics: In descriptive statistics, it deals with collection of data, its presentation in various forms, such as tables, graphs and diagrams and findings averages and other measures which would describe the data.For Example: Industrial statistics, population statistics, trade statistics etc Such as businessman make to use descriptive statistics in presenting their annual reports, final accounts, bank statements.(2) Inferential Statistics: In inferential statistics, it deals with techniques used for analysis of data, making the estimates and drawing conclusions from limited information taken on sample basis and testing the reliability of the estimates. For Example: Suppose we want to have an idea about the percentage of illiterates in our country. We take a sample from the population and find the proportion of illiterates in the sample. This sample proportion with the help of probability enables us to make some inferences about the population proportion. This study belongs to inferential statistics.

Classification of Data

The process of arranging data into homogenous group or classes according to some common characteristics present in the data is called classification.For Example: The process of sorting letters in a post office, the letters are classified according to the cities and further arranged according to streets.Bases of Classification: There are four important bases of classification:(1) Qualitative Base (2) Quantitative Base (3) Geographical Base (4) Chronological or Temporal Base (1) Qualitative Base: When the data are classified according to some quality or attributes such as sex, religion, literacy, intelligence etc(2) Quantitative Base: When the data are classified by quantitative characteristics like heights, weights, ages, income etc(3) Geographical Base: When the data are classified by geographical regions or location, like states, provinces, cities, countries etc(4) Chronological or Temporal Base: When the data are classified or arranged by their time of occurrence, such as years, months, weeks, days etc For Example: Time series data.Types of Classification:(1) One -way Classification: If we classify observed data keeping in view single characteristic, this type of classification is known as one-way classification.For Example: The population of world may be classified by religion as Muslim, Christians etc (2) Two -way Classification: If we consider two characteristics at a time in order to classify the observed data then we are doing two way classifications.For Example: The population of world may be classified by Religion and Sex.

(3) Multi -way Classification: We may consider more than two characteristics at a time to classify given data or observed data. In this way we deal in multi-way classification.For Example: The population of world may be classified by Religion, Sex and Literacy.

Construct a frequency distribution with suitable class interval size of marks obtained by students of a class are given below:23, 50, 38, 42, 63, 75, 12, 33, 26, 39, 35, 47, 43, 52, 56, 59, 64, 77, 15, 21, 51, 54, 72, 68, 36, 65, 52, 60, 27, 34, 47, 48, 55, 58, 59, 62, 51, 48, 50, 41, 57, 65, 54, 43, 56, 44, 30, 46, 67, 53

Solution: Arrange the marks in ascending order as12, 15, 21, 23, 26, 27, 30, 33, 34, 35, 36, 38, 39, 41, 42, 43, 43, 44, 46, 47, 47, 48, 48, 50, 50, 51, 51, 52, 52, 53, 54, 54, 55, 56, 56, 57, 58, 59, 59, 60, 62, 63, 64, 65, 65, 67, 68, 72, 75, 77 Minimum Value = Maximum = Range = Maximum Value Minimum Value = = Number of Classes = = = = = or approximate Class Interval Size () = = = or MarksClass LimitsC.LTallyMarksNumber ofStudents

ClassBoundaryC.BClassMarks

Note: For finding the class boundaries, we take half of the difference between lower class limit of the 2nd class and upper class limit of the 1st class. This value is subtracted from lower class limit and added in upper class limit to get the required class boundaries.

Frequency Distribution by Exclusive MethodClassBoundaryC.BTallyMarksFrequency

Geometric Mean

It is another measure of central tendency based on mathematical footing like arithmetic mean. Geometric mean can be defined in the following terms:Geometric mean is the nth positive root of the product of n positive given values Hence, geometric mean for a value containing values such as is denoted by of and given as under: (For Ungrouped Data) If we have a series of positive values with repeated values such as are repeated times respectively then geometric mean will becomes: (For Grouped Data) Where Example: Find the Geometric Mean of the values 10, 5, 15, 8, 12Solution: Here, and Example: Find the Geometric Mean of the following Data

Solution: We may write it as given below: Here, , , , , Using the formula of geometric mean for grouped data, geometric mean in this case will become: The method explained above for the calculation of geometric mean is useful when the numbers of values in given data are small in number and the facility of electronic calculator is available. When a set of data contains large number of values then we need an alternative way for computing geometric mean. The modified or alternative way of computing geometric mean is given as under:For Ungrouped DataFor Grouped Data

Example: Find the Geometric Mean of the values 10, 5, 15, 8, 12

Total

Example: Find the Geometric Mean for the following distribution of students marks:Marks

No. of Students

Solution: Marks No. of Students

Mid Points

Total

Harmonic Mean

Harmonic mean is another measure of central tendency and also based on mathematic footing like arithmetic mean and geometric mean. Like arithmetic mean and geometric mean, harmonic mean is also useful for quantitative data. Harmonic mean is defined in following terms:Harmonic mean is quotient of number of the given values and sum of the reciprocals of the given values.

Harmonic mean in mathematical terms is defined as follows: For Ungrouped DataFor Grouped Data

Example: Calculate the harmonic mean of the numbers: 13.5, 14.5, 14.8, 15.2 and 16.1Solution: The harmonic mean is calculated as below:

Total

Example: Given the following frequency distribution of first year students of a particular college. Calculate the Harmonic Mean. Age (Years)

Number of Students

Solution: The given distribution belongs to a grouped data and the variable involved is ages of first year students. While the number of students Represent frequencies.Ages (Years)

Number of Students

Total

Now we will find the Harmonic Mean as years.Example: Calculate the harmonic mean for the given below: Marks

Solution: The necessary calculations are given below:Marks

Total

Now we will find the Harmonic Mean as .

Example (4): The following data shows distance covered by persons to perform their routine jobs.Distance (Km)

Number of Persons

Calculate Arithmetic Mean by Step-Deviation Method; also explain why it is better than direct method in this particular case. Solution: The given distribution belongs to a grouped data and the variable involved is ages of distance covered. While the number of persons Represent frequencies.Distance Covered in (Km)Number of Persons

Mid Points

Total

Now we will find the Arithmetic Mean as Where , , and KmExplanation: Here from the mid points () it is very much clear that each mid point is multiple of and there is also a gap of from mid point to mid point i.e. class size or interval (). Keeping in view this, we should prefer to take method of Step-Deviation instead of Direct Method. Example (5): The following frequency distribution showing the marks obtained by students in statistics at a certain college. Find the arithmetic mean using (1) Direct Method (2) Short-Cut Method (3) Step-Deviation.Marks

Frequency

Solution: Direct MethodShort-CutMethodStep-DeviationMethod

Marks

Total

(1) Direct Method: or Marks(2) Short-Cut Method: Where Marks(3) Step-Deviation Method: Where MarksMerits and Demerits of Arithmetic Mean

Merits: It is rigidly defined. It is easy to calculate and simple to follow. It is based on all the abservations. It is determined for almost every kind of data. It is finite and not indefinite. It is readily put to algebraic treatment. It is least affected by fluctuations of sampling.Demerits: The arithmetic mean is highly affected by extreme values. It cannot average the ratios and precentages properly. It is not an approprite average for highly skewed distributions. It cannot be computed accurately if any item is missing. The mean sometimes does not coincide with any of the abserved value.

Merits and Demerits of Geometric Mean

Merits: It is rigidly defined and its value is a precise figure. It is based on all observations. It is capable of further algebraic treatment. It is not much affected by fluctuation of sampling. It is not affected by extreme values. Demerits: It cannot be calculated if any of the observation is zero or negative. Its calculation is rather difficult. It is not easy to understand. It may not coincide with any of the abservations.

Merits and Demerits of Harmonic Mean

Merits: It is based on all observations. It not much affected by the fluctuation of sampling. It is capable of algebraic treatment. It is an appropriate average for averaging ratios and rates. It does not give much weight to the large items. Demerits: Its calculation is difficult. It gives high weight-age to the small items. It cannot be calculated if any one of the items is zero. It is usually a value which does not exist in the given data.