**Notes of chapter: Statistics| Data Handling| Graphs are presented below. Indepth notes along with worksheets and NCERT Solutions.**

**(1) Statistics-**

The branch of the mathematics in which meaningful information is extracted from collected data is called **statistics.**

**Branches of Statistics –**

**(i)Descriptive statistics** –

The branch of statistics which describes, show or summarize data in a meaningful way is known as **descriptive statistics**. Data can be arranged in the form of tables, charts or graphs. We make conclusions according to our collected and analysed data.

**Eg:-** Collecting data to know the performance of the class.

**(ii) Inferential statistics** –

The branch of the statistics that is used to infer from the collected data to more general condition is known as** inferential statistics.**

**Eg:-** Results of exit poll is an example of inferential statistics.

**(iii)Survey sampling- **

The branch of the statistics that study the sample of population to get the required attribute of the population is known as **survey sampling.**

A population is the collection of data which we want to study.

**Eg:-**A survey is conducted by a company to know reason of declining of sale of their product.

**(2)Data-**

The facts or figures which are collected with a definite purpose are called **data.**

**Eg :-** Marks of students in class, temperature of cities etc.

**Types of Data**

**(i) Qualitative data-**

** Qualitative data** are those data which can not explain in numbers (Cardinal) but represents ordinal data such as name, colour, gender etc.

**Eg:-**

**Report card of a student**

Name of the student: Anu Rani

Class: viii B

Gender: Female

Language known; Hindi, English

Skill: Dance

Behaviour with class mates: Very good

Behaviour with teachers: Very good

Does she take interest in extra curricular activities: Yes

Does she play game: Yes

Name of the game: Badminton

Does she care for school property: Yes

**(ii)Quantitative Data-**

**Quantitative data **are those data which can explain in numbers (Cardinal) and can not represents ordinal data such as name, colour, gender etc.

**Eg:-**

Report card of a student quantitative data

Subject | Total Marks | Marks Obtained |

Hindi | 100 | 75 |

English | 100 | 80 |

Mathematics | 100 | 95 |

EVS | 100 | 85 |

Art | 100 | 70 |

General Knowledge | 100 | 75 |

Total | 600 | 480 |

**Types of quantitative data –**

**(a)Discrete data-**

**Discrete data** are those data which can be present only in whole numbers.

**Eg :-** Numbers like 23, 56, 78, 102 etc are discrete data.

**(b)Continuous data-**

**Continuous data** are those data which include the smallest part of a number.

**Eg:- **The height of any person can be 151cm or 151.5 cm.

**(3) Collection of data-**

Gathering information for required purpose is called **collection of data**.

**Eg:-** To know marks of class vii, we should collect the information regarding name of students, their roll numbers, subjects ,marks, maximum & minimum marks.

**Sources of data collection-**

Data can be collected by two sources.

**(i) Primary source –**

When data is collected by directly by researchers first time for the purpose of study is called **primary source.**

**Eg:-** Collecting data by questionnaire , interview or any other method.

**(ii)Secondary source: –**

When data is borrowed by any institution or organisation for required purpose is called **secondary source.**

**Eg:-** Data of world Bank used by researchers .

**(4)Organisation of Data**

Arranging collected data in systematic way to make study essay is called **organisation of data**.

**Methods to organise data**

**(i) Frequency distribution table**

**(ii) Charts or graphs**

**(iii) Calculate measures**

**Frequency**is a number that shows how many times a particular entry occurs in given data.

**Eg:-** In the above example 5, 4 and 2 are the frequencies of Cricket, badminton and tennis game respectively.

- When ungrouped data are presented in ascending or descending order, is called an
**array.**

**Eg:-** Marks of 10 students are given below:

55, 75, 87, 90, 74, 58, 98, 86, 66, 76

Prepare an array in ascending order.

**Ans-**

Array of given data arranging data in ascending order-

55,58, 66, 74, 75, 76, 86, 87, 90, 98

**(a) R****aw data or ungrouped data-**

When data are gathered in original form is called **raw data or ungrouped data.**

**Eg:-** Marks of 10 students are given below:

55, 75, 87, 90, 74, 58, 98, 86, 66, 76

These data are raw data or ungrouped data.

Frequency distribution table for the ungrouped data is known as **ungrouped frequency distribution data**.

**Eg:-** Marks obtained by 40 students of class VIII in Mathematics are given below:-

50, 53, 48, 60, 75, 86, 88, 99, 78, 70, 56, 100, 99, 95, 72, 84, 85, 89, 67, 69 80, 86, 84, 83, 67, 99, 95, 60, 75, 48, 99, 86, 95, 99, 84, 89, 77, 76, 75, 87

Frequency distribution table of above data is given below:-

**Method 2**

Step 1:-

Decide class interval. Any number can be taken as class interval. Generally 2,4 ,5 and 10 are taken as class interval. In this method, class interval should be taken according to data. If data is large class interval should be large, to shorten the frequency distribution table. If data are small class interval should be small to keep the size of the frequency distribution table in appropriate size.

In given example of ungrouped frequency distribution, we decide to take 10as class interval.

Method 2 is easy and frequently used by statisticians. First method should be used only if it is asked to do in your question paper. Otherwise, any method can be used.

**(ii)Presentation of data by charts or graphs**

**Graphical presentation of data-**

Data can be represented graphically by three different ways.

** (a) Bar Graphs-**

A **bar graph **is a representation of numbers using bars of uniform width. The length of the bars shows frequency.

Steps to draw bar graph:

Step 1

Show frequencies on the y axis. Choose a proper scale starting from zero. End the scale at a value greater than the highest value of given frequencies. Show these frequencies on the Y axis using equal divisions.

Step 2-

Show the variables on the X axis using equal width rectangles or bars. These bars have equal distance between them.

Step 3-

Bars length will show the number of frequency. Therefore, first mark frequency of variable with the help of Y axis scale and draw a line connecting it to x axis and draw a rectangle of equal width. Leave equal space between bars. To choose size of width first measure length of X-axis and then divide it by total of number of variables and space between them. It will give width of one bar. Or you can simply mark variable on X-axis and draw the bars.

**Eg:-** A bar graph is showing below:-

**Double bar graph**

**Double bar graph **shows two set of data with two adjacent bars with uniform width. The lengths of bars show different sets of data represent frequency of sets respectively.

Double bar can be drawn like bar graph.

**Eg:-** A double bar graph is showing below:-

**(b) Histograms of uniform width, and of varying width**

**Histograms of uniform width**

Histogram is a bar graph which is used to represent grouped data. Grouped data is represented on X – axis and frequencies are represented on Y – axis. The height of the bars shows the frequencies. There is no gap between the bars because there is no gap between the class intervals.

These graphs can be drawn like bar graph with no space between bars.

**Eg 1:-**

Group frequency distribution of marks of 60 students is given below. Represent these data graphically or draw a histogram for these data.

**Ans- Histogram of uniform width with given data is showing below:**

**Eg 2:-** Group frequency distribution of marks of 40 students is given below. Draw a histogram for these data.

**Ans-**In this example, class interval is not starting from 0. We will show data from 0 to 40 by a broken line.

Histogram of uniform width is showing below:

**Histograms of varying width**

When the class intervals of data are equal, the widths of the rectangles are equal. But, when class intervals of the data are not equal, the widths of the rectangles are varying. Therefore, the histogram does not give a correct picture. So, we have to take certain steps to make areas of the rectangles proportional to the frequencies.

**Eg :-** Draw a histogram of given data-

Marks | Number of students |

0 – 20 | 5 |

20 – 30 | 10 |

30 – 40 | 10 |

40 – 50 | 20 |

50 – above | 30 |

Total number of students | 75 |

**Ans-**

**Method 1 (With Histogram)**

First draw histogram and join mid points of rectangles. Assume that there is class interval before 0 – 10 and after 50 – 60 with zero frequency. Join midpoints of these class intervals to get frequency polygon. In given case, class interval lies in negative direction of X axis. The point where line segment of frequency polygon meets on vertical axis should be marked (A).

ABCDEFGH is required frequency polygon.

**Method 2(Without Histogram)**

Calculate midpoints of class intervals. Mark frequency corresponding to midpoints. Join all points. . Assume that there is class interval before 0 – 10 and after 50 – 60 with zero frequency. Join midpoints of these class intervals to get frequency polygon. In given case, class interval lies in negative direction of X axis. The point where line segment of frequency polygon meets on vertical axis should be marked (A).

The required frequency polygon is showing below:-

ABCDEFGH is required frequency polygon.

**Eg 2:-** Draw a frequency polygon for data given below:

Class marks |
Number of students |

40 – 50 | 2 |

50 – 60 | 3 |

60 – 70 | 5 |

70 – 80 | 8 |

80 – 90 | 13 |

100 – 110 | 1 |

Total | 40 |

**Ans-**

**Method 1 (With Histogram)**

First draw histogram and join mid points of rectangles. Assume that there is class interval before 40 – 50 and after 100 – 120 with zero frequency. Join midpoints of these class intervals to get frequency polygon. In given case, class interval lies in negative direction of X axis. The point where line segment of frequency polygon meets on vertical axis should be marked (A).

**Method 2(Without Histogram)**

Calculate midpoints of class intervals. Mark frequency corresponding to midpoints. Join all points. . Assume that there is class interval before 40 – 50 and after 100 – 110 with zero frequency. Join midpoints of these class intervals to get frequency polygon. In given case, class interval lies in negative direction of X axis. The point where line segment of frequency polygon meets on vertical axis should be marked (A).

Class marks |
Mid point of class interval |
Number of students |

40 – 50 | 45 | 2 |

50 – 60 | 55 | 3 |

60 – 70 | 65 | 5 |

70 – 80 | 75 | 8 |

80 – 90 | 85 | 13 |

100 – 110 | 95 | 1 |

Total | 40 |

**(d)** **Circle Graph or pie chart**

A** circle graph** shows the relationship between a whole and its parts .It is also known as** pie chart.**

**Eg:-**

The favourite games of the 100 students are given below:

Games | Number of students who like the game |

Cricket | 50 |

Badminton | 20 |

Football | 30 |

Draw a pie chart for given data.

**(e)** Line graph displays data that changes continuously over period of time and points represented particular data is joined by a straight line.

**Eg:-**

Time and temperature of Rita is tabulated in given table. Draw a line graph.

Time | 6 a.m. | 8 a.m. | 10 a.m. | 12 noon |

Temperature(^{0}c) |
37 | 35 | 40 | 37 |

**Ans-**

Step 1-

Horizontal line is x axis and time of given table is represented on it.

Mark points on x axis with equal distance and write values.

Step 2 –

Vertical line is y axis and temperature of given table is represented on it.

Mark points on y axis with equal distance and write values.

Step 3 –

Mark the points where value of time and temperature for the same time meets.

Step 4 –

Join all the points. The line is called line graph or time – temperature graph.

A line graph which is a whole unbroken line is called a **linear graph.** The linear graph is presented below:-

**(f) Coordinates** are values to identify exact position of a point on a graph.

**x – coordinate** represents or fixes the position of a point on x – axis.

**y – coordinate** represents or fixes the position of a point on y – axis.

Point A coordinates are (3, 4) where 3 is x – coordinate and 4 is y – coordinate.

Coordinates are discovered by mathematician Rene Descartes.

**Eg:-** Plot the point(2, 5) on the graph paper.

**Ans-**

Step 1 –

Draw x axis a nd y axis.

Step 2 –

Find 2 units on x axis and move 5 units up on y axis.

The point is required point.

**(iii)Representative values or central tendency of data**

Average is a number that represents the central tendency of a group of data or observations. Different forms of Data need different forms of central tendency to describe it. These are-

(i) Arithmetic Mean

(ii)Mode

(iii)Median

**Mode**

The mode of a set of observations is the observation that occurs most often.

**Eg:-** Find the mode of the given set of numbers:

1,2,3,5,6,6,7

Mode =6

**Helping Topics**