## Statistics: An Overview – CS Foundation Statistics Notes

Statistics is the study of the collection, organization, analysis, interpretation, and presentation of data. It deals with all aspects of data, including the planning of data collection in terms of the design of surveys and experiments. Statistics has perhaps its origin from Latin word ‘Status meaning political state’, word ‘Stato ‘ (Italian) and ‘Statisik’ (German) is also used for statistics. The word statistics is used in two senses

• Statistics data
• Statistical method

1. Statistics as numerical data
Webster defined Statistics as “The classified facts respecting the condition of the people in a state- especially those facts which can be stated in numbers or in tables of numbers or in any tabular or classified arrangement”.
The basic feature of statistics as quantitative or numerical data is as follows:

• Statistics are aggregates of facts.
• Statistics does not refer to a single figure but refers to a series of figures. Although the figure may be numerical if there is only one numerical figure it can not be compared. Thus for statistics, it is required that more than one figure be there.
• Statistics are numerically expressed.
• Statistics must have a numerical value. In the case of statistics data like old young, slow-fast are immaterial unless they have numerical data. This qualitative data is of no use for statistics.
• Statistics must be comparable to each other. Thus the figures collected should be comparable. It means the figures collected should be homogeneous for comparison and not heterogeneous.
• Statistics are affected to a marked extent by a multiplicity of causes.

2. Statistics as the statistical method
Bowley says “Statistics, may be called the science of counting”. At another place, he says that “Statistics may rightly be called the science of averages”.

3. Stages of statistical enquiry
Different stages of enquiry are

1. Planning the enquiry
Before collecting the data it is to predefine about the purpose, so that only the relevant data is collected The plan section is about how the data will be gathered.. After the purpose is decided, then it is to be decided what all data will be required

2. Collection of data
The collection part is the backbone of the inquiry. If the collection of data is not in proper form, in that case the conclusions drawn can never be reliable. The source of data may be primary i.e., collected directly, or the data may be secondary i.e., available from existing published sources. The first hand collection of data is one of the
most difficult and important tasks faced by file investigator. Primary data is the one that is collected directly from the source by investigator and secondary data is the one which is collected from already published one. Reasons for preferring a primary source

– The secondary source may contain mistakes due to errors which may not be easy to detect.
– The primary source usually includes definitions of terms and units used.
– The primary source often includes a copy of the schedule/questiortnaire and description of the procedures used in selecting the sample and in collecting the data.
– The primary source usually shows data in greater detail. The secondary source may omit some of the information.

3. Organisation of the data
The most important point in organizing a group of data is editing. This is done to correct omission, inconsistencies and wrong calculations in the survey. The classification is done to arrange the data according to some common characteristics. The last step in organisation is tabulation. The object of tabulation is to arrange the data in columns and rows for complete clarity.

4 Presentation of the organised data
After collection and organisation, the data should be presented. Finally the data is given the shape of diagram, chart and graph wherever necessary.

5 Analysing the presented data
Once the data are collected, organized and presented, the next step is that of analysis. The methods most commonly used are measures of Central Tendency and are called measures of the first order. Measures of Dispersion are called measures of the second order. Skewness, Correlation, Regression, Interpolation etc. are called measures of the third order. The analysis of facts based on observation is termed as (a) Scientific analysis, (b) Numerical analysis, (c) empirical analysis.

6 Interpretation of the collected data andforecasting
The last stage of statistical investigation is to derive the results and give comments on the inquiry in question. The interpretation of data is not an easy job and requires a high degree of skill and experience. If the analyzed data are not properly interpreted, the whole object of the inquiry may be erroneous. It is only correct interpretation which may lead to reliable conclusions.

4. Importance and functions of statistics
The various functions performed by statistics in modem times are discussed under:

1. Simplification of Complex Facts
The foremost purpose of the statistics is to simplify huge collection of numerical data. It is beyond the reach of human mind to remember and recollect the huge facts and figures. Statistical method makes it possible to understand the whole in the short span of time and in a better way.

2. Comparison
Comparison of data is yet another function of statistics, simplifying the data; it can be correlated or compared by a certain mathematical questions like averages, ratios, coefficients etc.
Some of the techniques used for comparison are averages, graphs, index numbers etc

3 . Relationship between Facts
Statistical methods are used to investigate the cause and effect relationship between two or more facts.

4. Formulation and Testing of Hypothesis
The most theoretical function of statistics is to test the various types of hypotheses and discover a new theory. For instance, by using appropriate statistical tools we can test the hypothesis of whether a particular coin is fair or not, whether Indian consumers are brand loyal etc.

5. Forecasting
Statistical methods are of great use to predict the future course of action of the phenomenon. It is only on the basis of statistical techniques like time series etc we can forecast.

6. Enrich the Individual Knowledge:
Statistical methods sharpen the faculty of rational thinking and reasoning of an individual. It is a master-key that solves the problems of mankind in every sphere of life.

7. Classification of Data:
Classification refers to a process of splitting up the data into certain parts which helps in the matters of comparison and interpretation of the various features of the data. This is done by the various improved ‘ techniques statistics.

8. To Draw Rational Conclusion:
In various fields of uncertainty like business and commerce, it is very much necessary to draw rational conclusions on the basis of facts collected and analyzed. For this, the mind of the decision-maker should be free from any bias and prejudices.

5. Statistics science or art
Statistics is science as it is the science of collecting, analyzing, presenting, and interpreting data. Governmental needs for census data as well as information about a variety of economic activities provided much of the early impetus for the field of statistics. Currently, the need to turn the large amounts of data available in many applied fields into useful information has stimulated both theoretical and practical developments in statistics.
Statistics can be termed as art because its application is dependent on the knowledge and skill of the person who is using it. It does not tell the facts but instead analyzes and gives us the merit and demerit and explains how to achieve the objective.

Actually, statistics is both sciences as well an art. Statistics is the art and science of deciding:

• what are the appropriate data to collect
• deciding how to collect them efficiently
• then use them to give information

6. Laws of statistics
There are two laws of statistics

1. Law of statistical regularity
2. Law of Inertia of a large number

1. Law of statistical regularity
The law of statistical regularity is the basic assumption in statistics that a random sample taken from a larger group will reflect the characteristics of the larger group. The larger the size of the sample in relation to the whole group, the more accurately it will reflect the group. This law holds good if the sample is large and is random.

2. Law of Inertia of large numbers
It is important because it guarantees stable long-term results for the averages of random events. The theorem is probability theory which states that the number of successes increases as the size of the sample increases.

7. Divisions of statistics

• Descriptive Statistics
Descriptive statistics are those which help reveal patterns in one or more sets of data through numerical analysis, Here the goal is to describe. Numerical measures are used to tell about features of a set of data. Descriptive statistics simply describe data. They include modes, means, ranges, and frequencies which help demonstrate striking trends, similarities, and differences within a set of data or between multiple sets of data. Descriptive statistics is concerned with telling about certain features of a data set. Although this is helpful in learning things as such it can not generalize.

• Inferential Statistics,
Inferential statistics refer to those used after analysis to make conclusions, evaluations, and further predictions based on previous research. Inferential statistics study a statistical sample, and from this analysis is able to say something about the data from which the sample came. Inferential statistics does start with a sample and then generalizes.

• Inductive statistics
The branch of statistics that deals with predictions, estimations, and decisions from data initially presented e.g. forecasting

• Mathematical statistics
It is the study of statistics from a mathematical standpoint, using other branches of mathematics such as linear algebra for analysis. Some of the techniques used are algebra, trigonometry, matrix theory, etc.

• Applied statistics,
This method is used to apply statistical methods to tackle a specific problem. Thus if we have to find out for which product there will be demand in the market we use applied statistics.

• Analytical statistics
A statistical study in which action will be taken on the basis of comparison.

8. Application of statistics
Statistics involves the collection, analysis and interpretation of data. A number of specialties have evolved to apply statistical theory and methods to various disciplines.
The areas where statistics can be applied are many

Statistics in business process applies statistical methods to data sets (often very large) to develop new insights and understanding of business performance & opportunities. Business statistics is the science of good decision making in the face of uncertainty and is used in many disciplines such as financial analysis, econometrics, auditing, production and operations including services improvement, and marketing research.

Thus statistics is able to answer the basic question of economics that is what to produce when to produce and how much to produce out of our limited resources.
Statistics play a major role in economics. On the basis of data economic problems are understood and with the help of various techniques of statistics time series analysis, forecasting etc. Analysis can be done so that on the basis of this analysis suitable policies can be drafted

2. Education
Statistics help to understand the education level in the country. We can analyse on the base of data how many students are going to school ?, how many children from rural area are going to school etc. and on this basis the policies can be drafted by government.

3. State
State authorities are always in need of statistics as it can help them in drafting policies for the government. With the help of data collected the real picture of the work and after results of various policies drafted is revealed. Government has its own agencies which collect data for the government. What are the best combinations of fertilizers, pesticides, etc are all replied with the help of statistics.

4. Social science
It has vast application in this field also. So much information like mortality rate, population control, birth rate, etc are of great importance for our economy and statistics is the answer to sought out such problems.

5. Management
Various functions of management planning, organizing, control, etc need an analytical approach. We need data to make such analyses and take decisions. In business, decisions can not be done on a hit and trial basis thus the techniques of statistics are used to get informed decisions.

6. Natural Science
It helps to find the causes for the diseases, how the disease did spread, how it was controlled, and various other things. In research work also it is of great importance.

9. Limitations of statistics

1. Statistics does not study qualitative phenomena:
Statistics deals with facts and figures. So the quality aspect of a variable or the subjective phenomenon falls out of the scope of statistics. For example, qualities like honesty, intelligence, etc. Which cannot be numerically expressed Can not form part of statistical data. This limits the scope of the subject.

2. Statistical laws are not exact:
Statistical laws are not exact as incase of natural sciences. These laws are true only on average. They hold good under certain conditions. They cannot be universally applied. So statistics has less practical utility.

3. Statistics does not study individuals:
Statistics deals with an aggregate of facts. Single figures are not statistics. This is considered to be a major handicap of statistics. It is the study of group data.

4. Statistics can be misused:
Statistics is mostly a tool of analysis. Statistical techniques are used to analyze and interpret the collected information in an inquiry. As it is, statistics are more dependent on the analyst and his experience. Statistical methods used by fewer expert hands will lead to inaccurate results. Here the fault is not of the data but with the person who makes wrong use of it. Statements supported by statistics are more appealing and are commonly believed. For this, statistics are often misused. Statistical methods rightly used are beneficial but if misused these become harmful.

Statistics: An Overview MCQ Questions

Question 1.
Organization of data does Not include
a. Editing
b. Classification
c. Collection
d. Tabulation
c. Collection

Question 2.
Statistics is essential to psychologists because
a. People differ a lot and in many respects.
b. It is compulsory
c. People’s experiences are much the
d. By using statistic psychologist can claim to be the cleverest researchers
a. People differ a lot and in many respects.

Question 3.
Descriptive statistics are:
b. About whether or not you can generalize from a sample.
c. The methods used to describe the essential characteristics of the data.
d. All of the things.
c. The methods used to describe the essential characteristics of the data.

Question 4.
A procedure used to select a sample of n objects from a population in such a way that each member of the population is chosen strictly by chance, each member of the population is equally likely to be chosen, and every possible sample of a given size, n, has the same chance of selection is known as
a. Statistics thinking
b. Inferential statistic
c. Descriptive
d. All of the above
d. All of the above

Question 5.
A specific characteristic of a population is known as an
a. sample
b. parameter
c. statistic
d. descriptive statistic
b. parameter

Question 6.
A specific characteristic of a sample is known as an
a. sample
b. parameter
c. statistic
d. information
c. statistic

Question 7.
Statistics was also termed as
a. The science of king
b. Science of statecraft
c. Political arithmetic
d. All of the above Data is
d. All of the above Data is

Question 8.
a. Collection of information about a person
b. Collection of information regarding the comparative things
c. Both a & b
d. None of the above
b. Collection of information regarding the comparative things

Question 9.
Both descriptive and inferential statistic is used to change data into information that in turn is converted into ____________ that lead to better decision making.
a. knowledge
b. a process
c. a forecast
d. a parameter
a. knowledge

Question 10.
An observed set of the population that has been selected for analysis is called
a. a sample
b. a process
c. a forecast
d. descriptive statistics
a. a sample

Question 11.
Keeping in view of all the statistical methods like tabulation etc. we can call statistics as
a. It’s application depends on the skill 7 experience of the statistician
b. It’s methods are systematic and have general application
c. Both a & b
d. None of the above
a. It’s application depends on the skill 7 experience of the statistician

Question 12.
As per the law of statistics given by King
a. If data is chosen randomly from a large group then that data will invariably show the characteristic of the group
b. The law developed by King is more correct if applied to a large group
c. Data sample which is to be taken for a large group should be systematic so that invariably it shows the characteristic of large group
d. Both a &b
d. Both a &b

Question 13.
Where conclusions are taken out on the basis of estimates regarding the population. It is _______ type of statistics
a. Inductive statistics
b. Descriptive statistics
c. Applied statistics
d. None of the above
d. None of the above

Question 14.
Inferential statistics is a process that involves all of the following EXCEPT
a. estimating a parameter
b. estimating a statistic
c. test a hypothesis
d. analyze relationships
b. estimating a statistic

Question 15.
Why do we use inferential statistics?
a. Inferential statistics are used to help us to compare the sample to the whole population.
b. Inferential statistics are used to help us to generalize from the sample to the whole population.
c. Inferential statistics are used to help us to show the difference between the sample and the whole population.
d. All of the above apply to the use of inferential statistics.
b. Inferential statistics are used to help us to generalize from the sample to the whole population.

Question 16.
Statistics is an indispensable function of the State as
a. Suitable policies are to be formed
b. It is dispensable if applied only to Central Government
c. Improve the working of the system
d. Both a & c
d. Both a & c

Question 17.
The allocation of limited resources in the most appropriate manner amongst unlimited ends is
a. Statistics
b. Economics
c. Market
d. All of the above
b. Economics

Question 18.
For studying the various national income components it is required that data should be collected on
a. Expenditure
b. Investments
c. Income
d. All of the above
d. All of the above

Question 19.
Business managers also need statistics for planning as
a. Decisions can not be made on a trial basis
b. Always systematically; organized data help in the formation of developing decision policies
c. Both a & b
d. None of the above
c. Both a & b

Question 20.
Natural science uses statistics
a. For finding the data related to disease
b. To find IQ
c. For studying fertility
d. All of the above
a. For finding the data related to disease

Question 21.
In statistics the facts
a. Should be expressed numerically
b. Should be related to other facts for comparison
c. Should be datum
d. Both a & b
d. Both a & b

Question 22.
Statistical data can be collected by
a. Actual counting
b. Estimation
c. Measurement
d. All of the above
d. All of the above

Question 23.
Statistical data of estimation should be
a. Able to draw meaningful conclusions
b. Have reasonable accuracy
c. Both a & b
d. None of the above
c. Both a & b

Question 24.
We use inferential statistics to generalize from our samples and descriptive statistics to describe our samples.
a. True
b. False
c Partially true
d None of the above
a. True

Question 25.
For data to be statistics it should Not
a. Be numerically expressed
b. Be estimated
c. Have predetermined purpose
d. None of the above
d. None of the above

Question 26.
You’ve been given a data group that contains 256 students’ first names’ last names’ grades in school, and six standardized test scores. Which of the following is the most appropriate way to I code the names of the students?
a. Sort the date on a first-name basis and assign consecutive numbers to each name beginning with 100.
b. Sort the data on last name basis and assign consecutive numbers to each name beginning with 001.
c. assigns random numbers to students’ names.
d. Sort the data on the basis of test results and assigning consecutive numbers to each rank starting with 001.
b. Sort the data on last name basis and assign consecutive numbers to each name beginning with 001.

Question 27.
The definition of statistics explaining statistics to be the collection, presentation, analysis, and interpretation of numerical data was given by
a. A.L. Bowley
b. Lovin
c. Hing
d. Coxton and Cowden
d. Coxton and Cowden

Question 28.
Secondary data is the data collected
a. By Investigation
b. From published data
c. From unpublished but collected data
d. Both b & c
d. Both b & c

Question 29.
Which of the following statistics would give you the estimate of the typical examination score of a class of 35 students?
a. A correlation coefficient
b. The variance
c. The standard deviation
d. The mean
d. The mean

Question 30.
What is a reasonable explanation for a data set of the test scores of 75 students in which the mean score is 81 and the median is 68?
a. An extremely low score
b. A number of typical scores
c. Too many scores
d. Two extremely high scores
d. Two extremely high scores

Question 31.
the study of the methods of organization summarization and presentation of statistical data is referred to as
a. inferential statistics
b. descriptive statistics
c. sampling
d. none of the above
b. descriptive statistics

Question 32.
The commonly found limitations to statistics are
a. It is dealing for individuals
b. Statistics gives precise and accurate information
c. Results of statistics are based on assumptions
d. All of the Above
c. Results of statistics are based on assumptions

Question 33.
Select the principles and methods used in statistics
a. Organization of data
b. Collection of data
c. Interpretation of data
d. All of the above
d. All of the above

Question 34.
Descriptive statistics is concerned with ______ while the goal of statistical inference is to ______ with the help of ________
a. generalizing sample data, draw inductive conclusions, probabilities
b. summarizing data, draw inductive conclusions probabilities
c. describing data, draw deductive concussions, probabilities
d. describing data, draw deductive conclusions, histograms
b. summarizing data, draw inductive conclusions probabilities

Question 35.
The only science that enables different experts using the same figures to draw different conclusions is called
a. Economics
b. philosophy
c. Statistics
d. Mathematics
c. Statistics
Hint
Statistics is the study of the collection, organization, analysis, interpretation and presentation of data. It deals with all aspects of data, including the planning of data collection in terms of the design of surveys and experiments.

Question 36.
The techniques that are used in summarizing and describing the main features of data constitute
a. Descriptive Statistics
b. Inductive Statistics
c. Mathematical Statistics
d. Applied Statistics
a. Descriptive Statistics
Hint
Descriptive statistics are those which help reveal patterns in one or more sets of data through numerical analysis, Here the goal is to describe. Numerical measures are used to tell about features of a set of data. Descriptive statistics simply describe data. They include modes, means, ranges and frequencies which help demonstrate striking trends, similarities and differences within a set of data or between 1 multiple sets of data.

Question 37.
One of the following does not qualifies to be called “statistics”
a. Tables of numbers
b. Census results for a number of districts
c. The daily number of cars passing through a traffic intersection
d. A time series of marks scored by a student in each year till class 12<sup>th</sup>
a. Tables of numbers
Hint
The basic feature of statistics as quantitative or numerical data are as follows:
– Statistics are aggregates of facts.
– Statistics does not refer to a single figure but it refers to a series of figures. Although the figure may be numerical if there is only one numerical figure it can not be compared. Thus for statistics, it is required that more than one figure be there.
– Statistics are numerically expressed.
– Statistics must have a numerical value. In the case of statistics data like old young, slow-fast are immaterial unless they have numerical data. This qualitative data is of no use for statistics.
– Statistics must be comparable to each other. Thus the figures collected should be comparable. It means the figures collected should be homogeneous for comparison and not heterogeneous.
– Statistics are affected to a marked extent by the multiplicity of causes.

Question 38.
Arrange the following steps in order of a statistical enquiry
(i) Interpretation of data
(ii) Analysis of data
(iii) Collection of data
(iv) Organisation of data
The options are
a. (a) (iii), (iv), (ii), (i)
b. (b) (iv), (iii), (ii), (i)
c. (C) (i), (ii), (iii), (iv)
d. (d) (ii), (iii), (iv), (i)
a. (a) (iii), (iv), (ii), (i)
Hint
Stages of statistical inquiry Different stages of enquiry are
1. Planning the enquiry
2. Collection of data
3. Organisation of the data
4. Presentation of the organised data-
5. Analysing the presented data-
6. Interpretation of the collected data and forecasting

Question 39.
Which of the following statements can be ranked as statistics?
a. India’s real per capita income in 2013 is ~ 50,000 whereas in China, Japan, South Korea and other ‘countries it is more than ~1,00,000
b. Neha is taller than Amrita
c. Abdul earns Rs. 5 crores for a single endorsement.
d. Today rainfall was estimated at 6 cm.
a. India’s real per capita income in 2013 is ~ 50,000 whereas in China, Japan, South Korea and other ‘countries it is more than ~1,00,000
Hint
The basic feature of statistics as quantitative or numerical data are as follows:
– Statistics are aggregates of facts.
– Statistics does not refer to a single figure but it refers to a series of figures. Although the figure may be numerical but if there is only one numerical figure it can not be compared. Thus for statistics, it is required that more than one figure be there .
– Statistics are numerically expressed.
– Statistics must have a numerical value. In case of statistics data like old young, slow-fast are immaterial unless they have numerical data. This qualitative data is of no use for statistics.
– Statistics must be comparable to each other. Thus the figures collected should be comparable. It means the figures collected should be homogeneous for comparison and not heterogeneous.
– Statistics are affected to a marked extent by the multiplicity of causes.

Question 40.
in the method, the sample is selected on a basis other than on probability considerations. Which one of the following is the correct option to fill in the blank space?
a. Non-random sampling
b. Stratified sampling
c. Systematic random sampling
d. Simple random sampling.
b. Stratified sampling
Hint
A type of probability sampling method in which sample members from a larger population are selected according to a random starting point and a fixed, periodic interval.

Question 41.
Arrange the following stages of a statistical inquiry in proper order:
(I) Organisation and presentation of data
(II) Interpretation of data
(III) Collection of data
(IV) Analysis of data
The correct option is –
a. (a) (III), (I), (IV), (II)
b. (b) (I), (II), (III), (IV)
c. (c) (IV), (III), (II), (I)
d. (d) (II), (I), (III), (IV)
a. (a) (III), (I), (IV), (II)
Hint
Stages of statistical enquiry Different stages of enquiry are
1. Planning the enquiry
2. Collection of data
3. Organisation of the data
4. Presentation of the organised data-
5. Analysing the presented data- .
6. Interpretation of the collected data and forecasting

Question 42.
Which of the following statements qualifies to be called ‘statistics’?
a. Metro rail length all over India is 600 Kms.
b. Indian automobile industry produced 2 million cars during the year 2012-13
c. The minimum support price for wheat for the crop season 2013-14 has been fixed at Rs.1,800 per quintal, as against Rs. 1,750 in the previous season
d. More people0 are making use of public transport, thereby causing traffic problems.
b. Indian automobile industry produced 2 million cars during the year 2012-13
Hint
The basic feature of statistics as quantitative or numerical data are as follows:

• Statistics are aggregates of facts.
• Statistics does not refer to a single figure but it refers to a series of figures. Although the figure may be numerical but if there is only one numerical figure it can not be compared. Thus for statistics, it is required that more than one figures be there .
• Statistics are numerically expressed.
• Statistics must have a numerical value. In the case of statistics data like old young, slow-fast are immaterial unless they have numerical data. This qualitative data is of no use for statistics.
• Statistics must be comparable to each other. Thus the figures collected should be comparable. It means the figures collected should be homogeneous for comparison and not heterogeneous.
• Statistics are affected to a marked extent by a multiplicity of causes.

Question 43.
The quote about the nature of statistics, “Statistics are like they clay from which you can make a God or a devil as you please”, is attributed to-
a. Croxton and Cowden
b. W. I. King
c. A. L. Bowley
d. Lincoln L. Chao.
b. W. I. King

Question 44.
Bio-statistics is a part of
a. Biology
b. Management
c. Social-Science
d. Economics
c. Social-Science

Question 45.
Which of the following can not be considered as a use of statistics in Business Management:
a. Fertility rate
b. Quality control
c. Forecasting
d. Production planning
b. Quality control

Question 46.
From the view of nature, statistics is:
a. An art
b. Neither science nor art
c. Science and art both
d. A science
c. Science and art both
Hint
Statistics is science as it is the science of collecting, analyzing, presenting, and interpreting data. Governmental needs for census data as well as information about a variety of economic activities provided much of the early impetus for the field of statistics. Statistics can be termed as art cause its application is dependent on the knowledge and skill of the person who is using it. Actually, statistics is both science as well as art.

Question 47.
The lower limit of class interval 20 – 30 is:
a. 25
b. 20
c. 30
d. 15
b. 20

Question 48.
“If a large sample is taken from a population in a random way, it will be fairly representative of the population”. This statement forms the:
a. Doesn’t form any law
b. Law of inertia of large numbers
c. Law of statistical regularity
d. Central limit theorem.
c. Law of statistical regularity
Hint
The law of statistical regularity is the basic assumption in statistics that a random sample taken from a larger group will reflect the characteristics of the larger group. The larger the size of the sample in relation to the whole group, the more accurately it will reflect the group This law holds good if the sample is large and is random.

Question 49.
Which of the following is not a function of statistics?
a. Giving Decisions
b. Testing Hypothesis
c. Framing Hypothesis
d. Facilitating comparisons.