Category: Management Notes

The Range of a distribution gives a measure of the width (or the spread) of the data values of the corresponding random variable. For example, if there are two random variables X and Y such that X corresponds to the age of human beings and Y corresponds to the age of turtles, we know from our general knowledge that the variable corresponding to the age of turtles should be larger.

Since the average age of humans is 50-60 years, while that of turtles is about 150-200 years; the values taken by the random variable Y are indeed spread out from 0 to at least 250 and above; while those of X will have a smaller range. Thus, qualitatively you’ve already understood what the Range of a distribution means. The mathematical formula for the same is given as:

Range=L–S

where L – the largets/maximum value attained by the random variable under consideration and S – the smallest/minimum value.

Properties

• The Range of a given distribution has the same units as the data points.
• If a random variable is transformed into a new random variable by a change of scale and a shift of origin as –

Y = aX + b

where Y – the new random variable, X – the original random variable and a,b – constants. Then the ranges of X and Y can be related as –

R_Y = |a|R_X

Clearly, the shift in origin doesn’t affect the shape of the distribution, and therefore its spread (or the width) remains unchanged. Only the scaling factor is important.

• For a grouped class distribution, the Range is defined as the difference between the two extreme class boundaries.
• A better measure of the spread of a distribution is the Coefficient of Range, given by:

Coefficient of Range (expressed as a percentage)=L–SL+S×100

Clearly, we need to take the ratio between the Range and the total (combined) extent of the distribution. Besides, since it is a ratio, it is dimensionless, and can, therefore, one can use it to compare the spreads of two or more different distributions as well.

• The range is an absolute measureof Dispersion of a distribution while the Coefficient of Range is a relative measure of dispersion.

Due to the consideration of only the end-points of a distribution, the Range never gives us any information about the shape of the distribution curve between the extreme points. Thus, we must move on to better measures of dispersion. One such quantity is Mean Deviation which is we are going to discuss now.

A simple way to define a harmonic mean is to call it the reciprocal of the arithmetic mean of the reciprocals of the observations. The most important criteria for it is that none of the observations should be zero.

A harmonic mean is used in averaging of ratios. The most common examples of ratios are that of speed and time, cost and unit of material, work and time etc. The harmonic mean (H.M.) of n observations is

H.M. = 1÷ (1⁄n ∑ _{i= 1}ⁿ (1⁄x_i) )

In the case of frequency distribution, a harmonic mean is given by

H.M. = 1÷ [1⁄N (∑ _{i= 1}ⁿ (f_i ⁄ x_i)], where N = ∑ _{i= 1}ⁿ f_i

Properties of Harmonic Mean

• If all the observation taken by a variable are constants, say k, then the harmonic mean of the observations is also k
• The harmonic mean has the least value when compared to the geometric mean and the arithmetic mean

Advantages of Harmonic Mean

• A harmonic mean is rigidly defined
• It is based upon all the observations
• The fluctuations of the observations do not affect the harmonic mean
• More weight is given to smaller items

Disadvantages of Harmonic Mean

• Not easily understandable
• Difficult to compute

A geometric mean is a mean or average which shows the central tendency of a set of numbers by using the product of their values. For a set of n observations, a geometric mean is the nth root of their product. The geometric mean G.M., for a set of numbers x₁, x₂, … , x_n is given as

G.M. = (x₁. x₂ … x_n)^1⁄n

or, G. M. = (π _{i = 1}ⁿ x_i) ^1⁄n = ⁿ√( x₁, x₂, … , x_n).

The geometric mean of two numbers, say x, and y is the square root of their product x×y. For three numbers, it will be the cube root of their products i.e., (x y z) ^1⁄3.

Properties of Geometric Means

• The logarithm of geometric mean is the arithmetic mean of the logarithms of given values
• If all the observations assumed by a variable are constants, say K >0, then the G.M. of the observation is also K
• The geometric mean of the ratio of two variables is the ratio of the geometric means of the two variables
• The geometric mean of the product of two variables is the product of their geometric means

Advantages of Geometric Mean

• A geometric mean is based upon all the observations
• It is rigidly defined
• The fluctuations of the observations do not affect the geometric mean
• It gives more weight to small items

Disadvantages of Geometric Mean

• A geometric mean is not easily understandable by a non-mathematical person
• If any of the observations is zero, the geometric mean becomes zero
• If any of the observation is negative, the geometric mean becomes imaginary

The arithmetic mean,’ mean or average is calculated by summ­ing all the individual observations or items of a sample and divid­ing this sum by the number of items in the sample. For example, as the result of a gas analysis in a respirometer an investigator obtains the following four readings of oxygen percentages:

14.9

10.8

12.3

23.3

___________

Sum=61.3

He calculates the mean oxygen percentage as the sum of the four items divided by the number of items—here, by four. Thus the average oxygen percentage is

Mean = 61.3 / 4 =15.325%

Calculating a mean presents us with the opportunity for learning statistical symbolism. An individual observation is symbo­lized by Y_i, which stands for the ith observation in the sample. Four observations could be written symbolically as Yi, Y₂, Y₃, Y₄.

We shall define n, the sample size, as the number of items in a sample. In this particular instance, the sample size n is 4. Thus, in a large sample, we can symbolize the array from the first to the nth item as follows: Y₁, Y₂…, Y_n. When we wish to sum items, we use the following notation:

The capital Greek sigma, Ʃ, simply means the sum of items indica­ted. The i = 1 means that the items should be summed, starting with the first one, and ending with the nth one as indicated by the i = n above the Ʃ. The subscript and superscript are necessary to indicate how many items should be summed. Below are seen increasing simplifications of the complete notation shown at the extreme left:

Properties of Arithmetic Mean:

1. The sum of deviations of the items from the arithmetic mean is always zero i.e.

∑(X–X) =0.

2. The Sum of the squared deviations of the items from A.M. is minimum, which is less than the sum of the squared deviations of the items from any other values.
3. If each item in the series is replaced by the mean, then the sum of these substitutions will be equal to the sum of the individual items.                       

Merits of A.M:

1. It is simple to understand and easy to calculate.
2. It is affected by the value of every item in the series.
3. It is rigidly defined.
4. It is capable of further algebraic treatment.
5. It is calculated value and not based on the position in the series.

Demerits of A.M:

1. It is affected by extreme items i.e., very small and very large items.
2. It can hardly be located by inspection.
3. In some cases A.M. does not represent the actual item. For example, average patients admitted in a hospital is 10.7 per day.
4. A.M. is not suitable in extremely asymmetrical distributions.

Tabulation is the systematic arrangement of the statistical data in columns or rows. It involves the orderly and systematic presentation of numerical data in a form designed to explain the problem under consideration. Tabulation helps in drawing the inference from the statistical figures.

Tabulation prepares the ground for analysis and interpretation. Therefore a suitable method must be decided carefully taking into account the scope and objects of the investigation, because it is very important part of the statistical methods.

Types of Tabulation

In general, the tabulation is classified in two parts, that is a simple tabulation, and a complex tabulation.

Simple tabulation, gives information regarding one or more independent questions. Complex tabulation gives information regarding two mutually dependent questions.

Two-Way Table

These types of table give information regarding two mutually dependent questions. For example, question is, how many millions of the persons are in the Divisions; the One-Way Table will give the answer. But if we want to know that in the population number, who are in the majority, male, or female. The Two-Way Tables will answer the question by giving the column for female and male. Thus the table showing the real picture of divisions sex wise is as under:

Three-Way Table

Three-Way Table gives information regarding three mutually dependent and inter-related questions.

For example, from one-way table, we get information about population, and from two-way table, we get information about the number of male and female available in various divisions. Now we can extend the same table to a three way table, by putting a question, “How many male and female are literate?” Thus the collected statistical data will show the following, three mutually dependent and inter-related questions:

1. Population in various division.
2. Their sex-wise distribution.
3. Their position of literacy.

Presentation of Data

Presentation of data is of utter importance nowadays. Afterall everything that’s pleasing to our eyes never fails to grab our attention. Presentation of data refers to an exhibition or putting up data in an attractive and useful manner such that it can be easily interpreted. The three main forms of presentation of data are:

1. Textual presentation
2. Data tables
3. Diagrammatic presentation

Textual Presentation

The discussion about the presentation of data starts off with it’s most raw and vague form which is the textual presentation. In such form of presentation, data is simply mentioned as mere text, that is generally in a paragraph. This is commonly used when the data is not very large.

This kind of representation is useful when we are looking to supplement qualitative statements with some data. For this purpose, the data should not be voluminously represented in tables or diagrams. It just has to be a statement that serves as a fitting evidence to our qualitative evidence and helps the reader to get an idea of the scale of a phenomenon.

For example, “the 2002 earthquake proved to be a mass murderer of humans. As many as 10,000 citizens have been reported dead”. The textual representation of data simply requires some intensive reading. This is because the quantitative statement just serves as an evidence of the qualitative statements and one has to go through the entire text before concluding anything.

Further, if the data under consideration is large then the text matter increases substantially. As a result, the reading process becomes more intensive, time-consuming and cumbersome.

Data Tables or Tabular Presentation

A table facilitates representation of even large amounts of data in an attractive, easy to read and organized manner. The data is organized in rows and columns. This is one of the most widely used forms of presentation of data since data tables are easy to construct and read.

Components of Data Tables

• Table Number: Each table should have a specific table number for ease of access and locating. This number can be readily mentioned anywhere which serves as a reference and leads us directly to the data mentioned in that particular table.
• Title: A table must contain a title that clearly tells the readers about the data it contains, time period of study, place of study and the nature of classification of data.
• Headnotes: A headnote further aids in the purpose of a title and displays more information about the table. Generally, headnotes present the units of data in brackets at the end of a table title.
• Stubs: These are titles of the rows in a table. Thus a stub display information about the data contained in a particular row.
• Caption: A caption is the title of a column in the data table. In fact, it is a counterpart if a stub and indicates the information contained in a column.
• Body or field: The body of a table is the content of a table in its entirety. Each item in a body is known as a ‘cell’.
• Footnotes: Footnotes are rarely used. In effect, they supplement the title of a table if required.
• Source: When using data obtained from a secondary source, this source has to be mentioned below the footnote.

Construction of Data Tables

There are many ways for construction of a good table. However, some basic ideas are:

• The title should be in accordance with the objective of study: The title of a table should provide a quick insight into the table.
• Comparison: If there might arise a need to compare any two rows or columns then these might be kept close to each other.
• Alternative location of stubs: If the rows in a data table are lengthy, then the stubs can be placed on the right-hand side of the table.
• Headings: Headings should be written in a singular form. For example, ‘good’ must be used instead of ‘goods’.
• Footnote: A footnote should be given only if needed.
• Size of columns: Size of columns must be uniform and symmetrical.
• Use of abbreviations: Headings and sub-headings should be free of abbreviations.
• Units:There should be a clear specification of units above the columns.

The Advantages of Tabular Presentation

• Ease of representation: A large amount of data can be easily confined in a data table. Evidently, it is the simplest form of data presentation.
• Ease of analysis: Data tables are frequently used for statistical analysis like calculation of central tendency, dispersion etc.
• Helps in comparison: In a data table, the rows and columns which are required to be compared can be placed next to each other. To point out, this facilitates comparison as it becomes easy to compare each value.
• Economical: Construction of a data table is fairly easy and presents the data in a manner which is really easy on the eyes of a reader. Moreover, it saves time as well as space.

Classification of Data and Tabular Presentation

Qualitative Classification

In this classification, data in a table is classified on the basis of qualitative attributes. In other words, if the data contained attributes that cannot be quantified like rural-urban, boys-girls etc. it can be identified as a qualitative classification of data.

Quantitative Classification

In quantitative classification, data is classified on basis of quantitative attributes.

Temporal Classification

Here data is classified according to time. Thus when data is mentioned with respect to different time frames, we term such a classification as temporal.

Spatial Classification

When data is classified according to a location, it becomes a spatial classification.

Advantages of Tabulation

1. The large mass of confusing data is easily reduced to reasonable form that is understandable to kind.
2. The data once arranged in a suitable form, gives the condition of the situation at a glance, or gives a bird eye view.
3. From the table it is easy to draw some reasonable conclusion or inferences.
4. Tables gave grounds for analysis of the data.
5. Errors, and omission if any are always detected in tabulation.

Therefore the importance of a carefully drawn table is vital for the preparation of data for analysis and interpretation.