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Frequency distribution is a method to represent raw data in the form from which one can easily understand the information contained in a raw data, where a table which plays the manner in which cumulative frequencies are distributed over various classes is called a cumulative frequency distribution.

A table which displays the manner in which cumulative frequencies are distributed over various classes.

(i) Runs in an ascending order: 

0, 0, 0, 0, 6, 15, 15, 15, 15, 26, 34, 34, 34, 34, 35, 39, 42, 42, 47, 48, 48, 53, 54, 64, 64, 64, 67, 71, 71, 82, 90, 124, 139. 

(ii) Highest Score = 139 

(iii) 3 times, he did not scored any run. 

(iv) He scored 3 centuries. 

(v) He scored more than 50 runs 12 times.

(i) 

(ii) The highest score is 100.

(iii) The lowest score is 37

(iv) Range = Highest score – Lowest score

= 100 – 37

= 63

(v) If the pass marks are 40 2 students are failed.

(vi) 8 people have scored 75 or more.

(vii) The observation between 50 and 60 that are not actually appeared are 51, 54, 57

(viii) Number of students that have scored less than 50 are 5

The numbers of runs scored by a player in 25 innings are

26, 35, 94, 48, 82, 105, 53, 0, 39, 42, 71, 0, 64, 15, 34, 15, 34, 67, 0, 42, 124, 84, 54, 48, 139, 64, 47 

(i) Runs in ascending order are 0, 0, 0, 15, 26, 34, 35, 39, 42, 42, 47, 48, 48, 53, 54, 64, 64, 67, 71, 82, 84, 94, 105, 124, 139. 

(ii) The highest number = 139 

(iii) The player did not score any run 3 times 

(iv) He scored 3 centuries 

(v) He scored more than 50 run 12 times

The number of runs scored by a player in 25 inning are:

26, 35, 94, 48, 82, 105, 53, 0, 39, 42, 71, 0, 64, 15, 34, 67, 0, 42, 124, 84, 54, 48, 139, 64, 47

(i) Runs in ascending order are:

0, 0, 0, 15, 26, 34, 35, 39, 42, 42, 47, 48, 48, 53, 54, 64, 64, 67, 71, 82, 84, 94, 105, 124, 139

(ii) The highest number = 139

(iii) The player did not scored any run 3 times.

(iv) He scored 3 centuries.

(v) He scored more than 50 runs 12 times.

Given,

Class size = 25

First class interval = 200 - 224

(i) Seven class intervals are:

200 - 240, 225 - 249, 250 - 274, 275 - 299, 300 - 324, 325 - 349, 350 - 374

(ii) Class marks of 200 - 224: \(\frac{200+224}{2}\) = \(\frac{424}{2}\) = 212

Class marks of 225 - 249 = \(\frac{225+249}{2}\) = \(\frac{474}{2}\) = 237

Class marks of 250 - 274 = \(\frac{250+274}{2}\) = \(\frac{524}{2}\) = 262

Class marks of 275 - 299 = \(\frac{275+299}{2}\) = \(\frac{574}{2}\) = 287

Class marks of 300 - 324 = \(\frac{300+324}{2}\) = \(\frac{624}{2}\) = 312

Class marks of 325 - 349 = \(\frac{325+349}{2}\) = \(\frac{674}{2}\) = 337

Class marks of 350 - 374 = \(\frac{350+374}{2}\) = \(\frac{724}{2}\) = 362

(i) Variable:

Any character that can vary from one individual to other is called variable.

(ii) Class - interval:

In the data of each group into which raw data is considered is called a Class - interval.

(iii) Class - size:

The difference between the upper class limit and the lower class limit is called the class size of the class.

(iv) Class - mark:

The middle value of the class is called the class-mark.Class mark = \(\frac{upper\,limit+lower\,limit}{2}\)

(v) Frequency:

The number of observations corresponding to class is called its frequency.

(vi) Class - limit:

Each class is bounded by two figures, called the class limits. The figures on the left side of the classes are called lower limits while the figures on the right sides are called the upper limits.

(vii) True class limits:

If classes are inclusive e.g., 15 - 19, 20 - 24, 25 - 29, 30 - 34

Then, true lower limit of class = Lower limit of class – 0.5

And, true upper limit of class = Upper limit of class + 0.5

e.g., True limits of class 15 - 19 are 14.5 - 19.5

The word ‘Data’ means information. Statistical data are of two types:

(i) Primary data:

When an investigator collects data himself with a definite plan or design in his/her mind is called Primary data.

(ii) Secondary data:

Data which are not originally collected rather obtained from published or unpublished sources are known as secondary data.

(i) Variate: Any character that can vary from one individual to another. 

(ii) Class-integral: In the data of each group into which raw data is considered. 

(iii) Class-size: The difference between the upper limit and lower limit. 

(iv) Class-mark: The middle value of the selected class. 

It can be calculated as : 

Class-mark = (Upper limit + Lower limit) /2 

(v) Frequency: The number of observations corresponding to class. 

(vi) Class limits: Each class is bounded by two figures, called the class limits. The figure on the right side are called upper limit while figure on the left side of the class are called lower limit. 

(vii)True class limits: 

If classes are inclusive. For example: 5-9, 10-14, 15-19 and so on 

Then, true lower limit of class = Lower limit of class – 0.5 

And, true upper limit of class = Upper limit of class + 0.5 

In this case, True limits of the class is 5-9 are 4.5 and 9.5 

But if classes are exclusive like 20-30, 30-40, 40-50 and so on, then class limits and true class limits are the same.

The data obtained in original form are called raw data. Raw data does not give any useful information and is rather confusing to mind. Data is grouped so that it becomes understandable can be interpreted. According to various characteristics groups are formed by us. After grouping the data, we are in position to make calculations of certain values which will help us in describing and analyzing the data.

The data obtained in original form is called raw data. It is difficult to get information from the raw data and is rather confusing sometime. Data is grouped so that it becomes understandable and can be interpreted. Grouped data is helpful to make calculations of certain values which will help in describing and analyzing the data.

(i) Primary data: When an investigator collects row data with a definite plan without using existed sources is called primary data. 

(ii) Secondary data: Data which is obtained from other published or unpublished sources is called secondary data. 

Secondary data are collected by an individual or an institution for some purpose and are used by someone else in another context. 

Primary data are reliable and relevant because it is original in character and are collected by some individuals or by research bodies.

The word data means information statistical data and of two types 

(i) Primary data, 

(ii) Secondary data 

(i) Primary data: when an investigator collects data himself with a definite plan or design in his (her) mind is called primary data 

(ii) Secondary data: data which are not originally collected rather obtained from published or unpublished sources are known as secondary data. Secondary data are collected by an individual or an institution for some purpose and are used by someone else in another context 

Primary data are reliable and relevant because they are original in character and are collected by some individuals or by research bodies. 

The word “Statistics” is used in both its singular as well as its plural senses. 

(i) In singular sense: Statistics may be defined as the science of collection, presentation, analysis and interpretation of numerical data 

(ii) In plural sense: Statistics means numerical facts or observations collected with definite purpose 

For example: Income and expenditure of persons in a particular locality, number males and females in a particular town are statistics.

Fundamental characteristics of statistics: 

1. A single observation does not form statistics are a sum total of observation 

2. Statics are expressed quantitatively not qualitatively 

3. Statistics are collected with definite purpose 

4. Statistics in an experiment are comparable and can be classified various groups. 

Below are some of the fundamental characteristics: 

(1) Statistics are collected with definite purpose. 

(2) Statistics are expressed quantitatively not qualitatively. 

(3) Statistics in an experiment are comparable and can be classified into different groups.

The word statistics is used in both its singular and plural senses. 

(i) In singular sense, statistics may be defined as science of collection, presentation, analysis and interpretation of numerical data. 

(ii) In plural sense, statistics means numerical facts or observations collected with definite purpose.

The word ‘Statistics’ is used in both singular as well as its plural sense.

(i) Statistics may be defined as the science of collection, presentation, analysis and interpretation of numerical data.

(ii) Statistics means numerical facts or observations collected with definite purpose.

The monthly pocket money of 6 friends is given below:

Rs. 45, Rs. 30, Rs. 40, Rs. 50, Rs. 25, Rs. 45

(i) Highest pocket money = Rs. 50

(ii) Lowest pocket money = Rs. 25

(iii) Range = Rs. 50 – Rs. 25

= Rs. 25

(iv) The amount of pocket money in ascending order is:

Rs. 25, Rs. 30, Rs. 40, Rs. 45, Rs. 45, Rs. 50

The monthly pocket money of six friends is given below: 

Rs 45, Rs 30, Rs 40, Rs 50, Rs 25, Rs 45 

(i) Highest pocket money = Rs 50 

(ii) Lowest pocket money = Rs 25 

(iii) Range = Highest pocket money – Lowest pocket money 

= Rs 50 – Rs 25 

= Rs 25 

(iv) Ascending order is: Rs 25, Rs 30, Rs 40, Rs 45, Rs 45,Rs 50.

(i) The arrangement according to the highest weight is:

3.1, 3.0, 2.9, 2.8, 2.7, 2.7, 2.6, 2.5, 2.4, 2.4, 2.3, 2.2, 2.1

(ii) The highest weight is 3.1 kg.

(iii) The lowest weight is 2.1 kg.

(iv) Range = Highest weight – Lowest weight

= 3.1 – 2.1

= 1.0 kg

(v) Number of babies that were born on that day were 15.

(vi) 4 babies weigh below 2.5 kg.

(vii) 4 babies weigh more than 2.8 kg.

(viii) 2 babies weigh 2.8 kg.

The weights of new born babies (in kg) area as follows 

2.3, 2.2, 2.1, 2.7, 2.6, 3.0, 2.5, 2.9, 2.8, 3.1, 2.5, 2.8, 2.7, 2.9, 2.4

(i) The weights in descending order

3.1, 3.0, 2.9, 2.9, 2.8, 2.8, 2.7, 2.7, 2.6, 2.5, 2.5, 2.4, 2.3, 2.2, 2.1

(ii) The highest weight = 3.1 kg 

(iii) The lowest weight = 2.1 kg 

(iv) Range = 3.1 - 2.1 = 1.0 kg

(v) 15 babies were born on that particular day 

(vi) 4 babies weight below 2.5 kg 

(vii) Weight more than 2.8 kg are 4-babies. 

(viii) Weight 2.8 kg --> 2 babies

Cumulative frequency distribution: A table which displays the manner in which cumulative frequencies are distributed over various classes is called a cumulative frequencies (or) cumulative frequencies distribution table.

The ages of ten students of a group are given below

8-6, 9-0, 8-4, 9-3, 7-8, 8-11, 8-7, 9-2, 7-10, 8-8

(i) Lowest age is 7 years, 8 months 

(ii) Highest age is 9 years, 3 months. 

(iii) Range = highest age – lowest age = 9 year, 3 months, 7 year, 8 months = 1 year, 7 months