16.1 Definition of Statistics

Originally the word “STATISTICS” was derived from the word “STATUS” or “STATE” ie, a science of dealing with the affairs of administration of a state. Today, statistics refers to the scientific approach to the collection, presentation, analysis, and interpretation of numerical data or information by presenting it in a way to understand the information.

Different individuals react differently under the same set of circumstances. But the ability to predict with reliable confidence how the entire group is likely to react under the given circumstances is very important. Hence, it is necessary to evaluate a certain statistical parameter, which would represent the characteristics of the entire group. This is called an average. The method of statistics actually deals with the evaluation of such characteristics and hence it is also defined as the “Science of Averages.”

16.2 Role of Statistics in Analysis

Today statistics is indispensable for a clear appreciation of any problem affecting human welfare, whether it is industry, transport, business, medicine, and more significantly, in quality control analysis.

Statistics is an aid to the economic measure. It is a technique of analyzing the data obtained to conclude on the economic progress and forecast the future trend.

Statistics is an aid to the manager particularly in these days of impersonal relationships between the employer and employee. With this, he can estimate the demand more accurately then by guesswork. Statistics help in recording the past knowledge and experience, drawing out standards whose results can be compared from time to time. And also the expected changes, the reasons for these changes, and the effects of these changes on the nation’s economy can be estimated.

We can summarize the role of statistics as follows:

1. Helps in simplifying complexity.

2. Keeps a control on the disturbing factors affecting the data and helps in detecting and eliminating them.

3. Studies the relationship between connected facts and in measuring their degree or level of significance.

4. Helps in forecasting the future happening of events based on the present situation.

16.3 Limitation of Statistics

1. It is restricted only to the study of quantitative phenomena.

2. It is cognizant of individual items.

3. It does not reveal the entire problem.

4. Its laws are true only on an average.

5. It is liable to be mishandled and misused.

16.4 Elements of Statistical Techniques

In general, the elements of statistical techniques include:

1. Collection of data.

2. Assembling of data.

3. Classification and summarization of data.

4. Presentation of data.

● in textual form

● in tabular form

● in graphic form

5. Analysis of data.

16.5 Methods of Collecting Data

The first step in statistical analysis is to collect data, which can be in any of the following methods:

A. By direct observation like counting the number of buses passing a particular junction during peak hours, or taking the time study of an operation.
Advantages: It reduces the chance of incorrect data or guess works being recorded.
Disadvantages: Many times this method is costly, or not possible, such as observing the various activities done by a housewife during the course of a month.

B. Interviewing asking concerned people personally for the required information, such as the market research people going around to the customers’ houses to collect information for the census data.
Advantages: Easy and more accurate replies.
Disadvantages:

(a) Sometimes deliberately wrong data may be given due to shyness or forgetfulness.

(b) Wrong understanding of the questions or wrong interpretation of the answer.

C. Referring to published records: They can be of two types:

(a) The primary data, consisting of raw and unprocessed data collected at the point of generation; and

(b) The secondary data, consisting of processed summarized primary data as found in reports and publications, either government or private.

D. Questionnaires: Sending questionnaires by post and asking people to complete it and send it back.
Advantages: Reduces wrong interpretation and giving all unnecessary information.
Disadvantages: Poor response. Just out of laziness, namely a maximum number of 15% respond. Also by the time you get back the answers, it might be too late.

Factors for the Design of a Questionnaire:

(a) Questions should be simple.

(b) Questions should not be vague or ambiguous.

(c) The questionnaire should be as short as possible.

(d) The questions should not be irrelevant or pursuant.

(e) Leading questions should not be asked. For example “Don’t you think all the sensible people use XYZ Soap”?

(f) Questions in a questionnaire should fall into a logical sequence.

(g) The best kinds of questions are those which allow preprinted multiple choice answers, so that the respondent has just to tick his answer.

16.6 Data Classification

To make the data comprehensive, it is necessary to classify it into homogeneous groups, subgroups, etc., as per their respective characteristics, into useful and logical categories.

The main objects of classification are:

1. To simplify the understanding of such huge data.

2. To enable specifying the main objective of collecting such figures.

3. To trace out certain characteristics in the order of their importance.

4. To facilitate comparison.

16.7 Data Presentation

The next step after data classification is to arrange them in an orderly manner, so that they can be readily understood. In general, the following are the different methods of data presentations.

1. Narrative form.

2. Tabular presentations.

3. Single dimensional diagrams, such as bar charts.

4. Two-dimensional presentations.

● Squares and rectangles.

● Graphs.

● Binominal curves.

● “Z” charts.

● Lorenz curves.

5. Pictorial presentations.

● Pictograms.

● Pie charts.

● Statistical maps.

6. 3-Dimensional presentations.

16.8 Population Versus Sample

Before understanding the elements of statistical techniques, the first step in statistical analysis is to understand the basic data groups like population, sample, attributes, variables, etc.

16.9 Attributes and Variables

Attribute: If the measurement is made qualitatively, then it is called an attribute. eg, color, sex, etc.

Variable: If the measurement is made quantitatively, then it is called a Variable. eg, height, weight, etc.

Discrete variable: If a variable can take only a specified number of values, then it is called a discrete variable.

Continuous variable: If a variable can take all the possible values between two real numbers, then it is called continuous variable.

16.10 Graphs

Graphs are visual presentation of the data which give an immediate visual concept of the trend or comparison.

A graph is a representation of data by a continuous curve on squared paper, while a diagram is any other two-dimensional form of visual representation. Note that a line on a graph is always referred to as a curve—even though it may be straight.

16.12 Innovative Graphs

These days, you find in newspapers statistics data represented in innovative types of graphs, a recent one noticed in the Times of India on the urban planning statistics of various cities/towns of India can be illustrated as in Fig. 16.9.

16.13 Frequency Graphs

These frequency graphs can be drawn in several forms.

16.13.3 Frequency Curve

This is obtained by smoothing the frequency polygon to obtain a curve shown in Fig. 16.10.

16.14 Ogive

It is the frequency polygon of the cumulative frequencies. Thus, an ogive is obtained by plotting the cumulative frequencies corresponding to values of the variate, to which they belong. Cumulative frequency polygon when smoothed out will be referred to as an ogive curve is shown in Fig. 16.11.

16.15 “Z” Chart

A “Z” chart (Fig. 16.13) is a combination of three graphs drawn for data over a period of one year incorporating:

(a) Individual monthly figures.

(b) Monthly cumulative figures for the year.

(c) A moving annual total.

Construction of Z-charts:

1. The basic curve representing the values of a variable over each of the 12 months of a particular year is drawn in a time scale (curve A).

2. The cumulative values for each month is computed and represented by curve B.

3. The total value of the variable for the 12 months, including that month plus the preceding 11 months, viz February of the previous year to January of this year is computed and indicated against January month. Similarly, the total of the variable from March of the previous year to February of this year is indicated against February. In this manner, the total of the 12 months’ values for the particular month plus the preceding 11 months is indicated against each of the months. A line joining these points is called the moving annual total curve (curve C) and since these three lines together look like Z of the English alphabet, it is called z-chart as illustrated in Fig. 16.12.

16.16 Lorenz Curves

It is a well-known fact that in practically every country, a small proportion of the population owns a large proportion of the total wealth. Industrialists know, too, that a small proportion of all the factories employ a large proportion of the factory workers. This disparity of proportions, which is similar to ABC analysis and Pareto principle, is a common economic phenomenon. American economist Max Lorenz illustrated this in 1905 by a graph called the Lorenz curve, showing the reality of wealth distribution. A straight diagonal line representing perfect equality of wealth distribution is drawn above it for providing contrast. The difference between the straight line and the curved line is the amount of inequality of wealth distribution, a figure also called the Gini coefficient. Lorenz curve is shown in Fig. 16.13.

16.17 Frequency Distribution

In earlier paragraphs, we have studied a variety of graphic methods of presentation of data. One of the frequent problems related to statistics is to represent grouped data graphically. Consider the following example (Table 16.1), indicating the daily income distribution among 50 workers, classified into 10 groups between Rs. 80 and Rs. 129.

16.18 Central Tendency

This frequency distribution table is the basic data representation for further computations and statistical analysis. The following few paragraphs explain the subsequent steps in analyzing the central tendency of the figures.

As a second step, this distribution can be represented as frequency polygon illustrated in Fig. 16.14. Where we join the top values of each income group, we get what is called the frequency distribution curve. This is also illustrated in Fig. 16.14, which indicates other parameters of central tendency.

You can see from the histogram of Fig. 16.14 that while the income group value (or x value) increases, the frequency (or y value) slowly increases from a low value and attains a high value at somewhere about the center of the value and again decreases towards the end of value. This is called the Central tendency, and is common in most statistical distribution.

16.19 Measures of Central Tendency

The following are some of the measures for the frequency distribution.

1. Mean (or average)

2. Median

3. Mode

Also the other measures for the dispersion of the data (ie, how the individual values are spread out) are

1. Range

2. Mean deviation

3. Standard deviation or variance

4. Kurtosis

5. Skewness

The three measures mean, median, and mode all sound similar, but the frequency curve as illustrated in Fig. 16.15, which has a deviation from the normal curve, explains their differences better.

Range, represented by d is the difference between the minimum and maximum values in a series.

Mean, represented by a, is the average point on either side of which the frequencies are equally distributed, that is around which the areas on either side are equal.

Median, represented by b, is the mid-point of the range, that is the point midway between the smallest and largest value of the frequency. The median is the value of the middle item when the items are arranged according to size.

Mode, represented by c, is the point corresponding to the largest value in the array, that is the peak of the frequency distribution.

16.20 Mean or an Average

Mean or an average is a typical value which is intended to sum up or describe the mass of data. It also serves as a basis for measuring or evaluating extreme or unusual values. The average is a measure of the location of the point of central tendency. Mean can be in four types as below:

(a) Arithmetic mean

(b) Geometric mean

(c) Harmonic mean

(d) Quadratic mean

16.21 Arithmetic Mean

Due to ease of computation and long usage, the arithmetic mean is the best known and most commonly used of all the averages. When the word “mean” is used without qualification, the reference is to the arithmetic mean.

Calculation of arithmetic mean: The arithmetic mean of a small group of individual values may be obtained by dividing the sum of the values by the number of items used.

The computation of the arithmetic mean is expressed in formula form as:

$\overline{X}=\frac{\Sigma \left(X\right)}{N}$

where

$\overline{X}$ = arithmetic mean.

∑ = symbol meaning “sum of.”

X = data expressed as individual values.

N = number of items.

16.22 Geometric Mean, Quadratic Mean, and Harmonic Mean

The three other means viz., the Geometric mean, Quadratic mean, and Harmonic mean are rarely used in statistics and hence, not covered here. However, for academic interest, the formulae are given below:

If X₁, X₂, X₃ … etc., are the X values, then

Geometric Mean = $\sqrt[n]{X_1\times X_2\times X_3\times X_4}$.

Quadratic Mean = $\sqrt{\frac{X^2}{N}}$.

Harmonic Mean is given by $\frac{1}{HM}=\frac{1}{X_1}+\frac{1}{X_2}+\frac{1}{X_3}+\frac{1}{X_4}\dots \frac{1}{X_n}$.

16.25 Dispersion

The average or typical value has little use unless the degree of variation which occurs about it is given. For if the scatter about the measure of central tendency is very large, the average is not a typical value. It is therefore necessary to develop a quantitative measure of the dispersion (or variation, or scatter) of values about the average.

16.26 Range

The range, the simplest of the measure of dispersion, and as defined earlier, is the difference between the minimum and maximum values in a series. It is sometimes given in the form of a statement of the minimum and maximum values themselves.

The difference between the two extreme values indicative of the spread of the series, but quite frequently is misleading, because it gives no information about how the items are dispersed.

16.27 Mean Deviation

The range is dependent for its value entirely upon the two extreme values. Obviously, when these end-values are far removed from the remainder of the data, a satisfactory measure of dispersion must be dependent upon the position of every value in the series.

A simple method for determining the scatter of a series of values about a given point is to take the average distance of the items from the given point. The smaller the average distance about this point, the smaller the scatter or dispersion of the values.

In a frequency distribution, the average distances of the items from the measure of control tendency, such as the arithmetic mean, may be used for this purpose. However, since the sum of the deviations about the arithmetic mean is zero, it is necessary to ignore signs in order to obtain the average distance of items from that measure.

16.28 Standard Deviation

The standard deviation represented by the Greek letter sigma, σ, is the most useful value in statistics and in total quality management to understand the deviation of the values from the mean in a distribution. It is computed by taking the quadratic mean of the deviations from the arithmetic mean of those values. A standard deviation close to 0 indicates that all values are close to the mean with a steep curve of high kurtosis. The lower the σ, the wider the variation. While the standard deviation is thus called the root-mean-square, the analysis of the deviation is called analysis of variation or ANOVA in short.

${\sigma }^2=\frac{{{{\displaystyle \sum }}^{\text{}}}^{\text{}}\left(X^2\right)}{N}$

where,

σ = standard deviation.

X = deviation of individual item from arithmetic mean = $X_n–\overline{X}$.

N = total number of items.

16.29 Skewness

Referring to Fig. 16.15, skewness is a term for the degree of distortion from symmetry exhibited by a frequency distribution.

When a distribution is perfectly symmetrical with one mode, the values of the mean, median, and mode coincide. In an asymmetrical (skewed) distribution, their values will be different and it will be farthest from the mode. The mode is not affected at all by unusual values; therefore the greater the degree of skewness the greater the distance between the mean and the mode.

Measure for skewness:

$S_k=\frac{\left(\text{Mean}-\text{Node}\right)}{\text{Standard}\text{Deviation}}$

16.30 Kurtosis

This is a measure of the peakness of the distribution as is clear from Fig. 16.16.

16.31 Conclusion

It can be seen that the understanding of basic statistics and the related terminology is a must for practicing statistical quality control, without which total quality management has no place.