CBSE Class 11 Maths Revision Notes Chapter 15

Class 11 Mathematics Revision Notes for Statistics of Chapter 15

Mathematics is a subject that requires students to grasp complex formulas and apply them to solve problems. One such material that they can depend on is revision notes as they contain step-by-step points for every topic. Especially for the chapter on Statistics in Class 11 Mathematics, students can refer to Class 11 Mathematics Chapter 15 Notes. These notes are provided by subject matter experts with accurate points that will help students with quick revisions. 

Class 11 Mathematics Revision Notes for Statistics of Chapter 15 – Free Download

Notes of Statistics Class 11 Statistics–Brief Chapter Overview

Terminologies

Statistics: Statistics is the study of gathering, organising, presenting, analysing, and interpreting numerical data.

Limit of the class: A class’s limitations are its maximum or minimum values. The figure that falls between the upper and lower limits is respectively the highest and lowest.

Class interval – The difference between the upper and lower limits of each class is known as the class interval.

Primary and secondary data – Data that is gathered by the investigator is referred to as primary data, while data gathered by someone else is referred to as secondary data.

Variable or variate – A variable or variate is a symbol or feature whose magnitude varies from observation to observation. Such as size, height, etc.

Frequency – The frequency of observation is the number of times it appears in a given set of data.

Discrete frequency distribution–A discrete frequency distribution is a frequency distribution in which the data is displayed in such a way that the precise measurements of the units are readily apparent.

Continuous frequency distribution– Continuous frequency distribution is the type of frequency distribution where the classes and groups are not perfectly measurable.

Cumulative frequency distribution–The frequency that is obtained after adding the frequencies of the first class, the second class, the third class, and so on is known as the cumulative frequency distribution. The provided frequencies ought to be organised into groups or classes.

Graphical Representation Of Frequency Distribution:

Histogram – To create a histogram, mark the corresponding frequencies on the y-axis and the given class intervals on the x-axis. An erected rectangle with a width proportional to the class interval and a length proportional to the frequency of that class interval is drawn in the corresponding intervals. The area of the rectangle is indicated by the frequency of that class when the class interval is used as the graph’s unit length.

The height of the rectangles is proportional to the ratio of the frequencies to the width of each class when the class intervals have different widths.

Bar diagrams – Only the length of the bars or rectangles is considered when drawing bar diagrams. The data is divided into various classes, which are then represented on the x-axis with equal widths and represented on the y-axis, with frequencies that are proportional to the length of each bar.

Pie diagrams – These show relative frequency distribution by dividing a circle into equal-sized sectors for each class, with the area of each sector corresponding to the frequency of that class. By proportionally dividing the angles according to the frequencies, the division is carried out.

Frequency Polygon – The variate values are plotted on the x-axis, and the corresponding frequencies are plotted on the y-axis, ‌ to draw the frequency polygon of an ungrouped frequency distribution. A trend is then shown by connecting the midpoints of each bar with a straight line

Cumulative frequency curve (Ogive) – We first make a cumulative frequency table using the supplied data. The lower or higher boundaries of the associated class intervals are then displayed against the cumulative frequencies, respectively. After the points are linked, the resulting curve is referred to as an ogive or cumulative frequency curve. ‘Ogives’ can normally be drawn in two different ways:

1. The “less than” method involves plotting the upper limits of the points on the x-axis and the corresponding less than cumulative frequencies on the ordinates or y-axis. The points are then joined free-hand to create an oblique or smooth curve. It is also known as a falling curve.
2. The “more than” method involves plotting the points with the lower limits on the x-axis and the corresponding more than cumulative frequencies on the ordinates or y-axis. The points are then joined free-hand to create an oblique or smooth curve. It is also known as a falling curve.

Measures Of Central Tendency:

The process of describing an entire data set using the central value of that data set is known as the measure of central tendency. The following list includes the five central tendency measures:

The ratio of the sum of the values of the items in a series to the total amount of data is known as the arithmetic mean.

• They are further of five types;

1. i) Arithmetic mean for unclassified data
2. ii) Arithmetic mean for frequency distribution 

iii) Arithmetic mean for classified data

1. iv) Combined mean
2. v) Weighted arithmetic mean 

Arithmetic Mean Properties –

1. a) Changes in scale and origin are never affected by arithmetic means.
2. b) A set of values with their arithmetic mean to have zero deviations when added together algebraically.
3. c) The sum of the squares of the deviations of a set of values is minimum when taken about the mean.

Symmetrical And Skew Distribution:

If the same number of frequencies is distributed on either side of the mode, the distribution is said to be symmetric. The frequency curve in this instance is bell-shaped and A=Md=Mo.

In the case of an anti-symmetric or skew distribution, the variation lacks symmetry. Assuming two situations,

• Positive skewness – The frequencies rise sharply at first and then gradually decline after reaching the modal value and A>Md>Mo.
• Negative skewness – The frequencies gradually increase at first before slowly declining after the modal value and A<Md<Mo.

Measure Of Dispersion:

Dispersion of the data is the degree to which numerical data tend to spread about an average value. There are three dispersions, enlisted below:

• The range is used to denote the difference between the highest and the lowest element of a data. It can be represented as:
• range=xmax−xmin.

The coefficient of range is expressed as xmax−xminxmax+xmin.

It finds its uses in statistics, especially in series relating to quality control in production.

1. i) Interquartile range is Q3−Q1.
2. ii) Semi-inter quartile range (quartile deviation) is QD = Q3−Q12

iii) Coefficient of quartile deviation is Q3−Q1Q3+Q1

1. iv) QD=2/3 SD

Mean Deviation is defined as the arithmetic mean of absolute deviations of the values of the variable from a measure of their average, which can be either of mean, median, or mode.

δ is used to denote it. The formula for various conditions is given below:

For simple (discrete) distribution δ= ∑ |x−z|n where n is the number of terms and z can be either of A or Md or Mo.

For unclassified frequency distribution δ=  ∑|x−z|∑f

For classified distribution, δ=  ∑|x−z|∑f x is for the class mark of the interval.

MD= 45SD

Average (mean or Median or Mode) = mean deviation from the averageaverage

Coefficient of Mean Deviation is the ratio of MD and the mean from which the deviation is measured and is given by MD= ∑|x−x¯¯¯|n

Analysis of Frequency Distributions:

• The measure of variability is called the coefficient of variation. It is independent of units. The letters C.V. are used for denoting it. It is defined as C.V= σx¯ ×100 where the term σx¯ is called the coefficient of standard deviation.
• The distribution for which the coefficient of variation is less is called more consistent.
• For two series with equal means, the series with a greater standard deviation is called more variable than the other. 
• The series with a lesser value of standard deviation is said to be more consistent than the other.

Notes of Statistics Class 11 Statistics – Brief Chapter Overview

Class 11 Mathematics Revision Notes Chapter 15

Students need not fret about their mathematics exam revision with the Class 11 Mathematics Chapter 15 Notes. They can easily access them from the Extramarks’ website as and when it is needed. The Class 11 Mathematics Notes Chapter 15 is written clearly and concisely for maximum learning and understanding of concepts included in Class 11 Mathematics. 

Revision Notes Class 11 Mathematics Chapter 15

Definition of Statistics and Some Useful Terms 

Statistics – It is used to gather, arrange, and present numerical data for its analysis and interpretation. 

The following are some crucial terms related to statistics:

Limit of the Class – The smallest and largest data values that can be included in a class are referred to as the class limit.

Class Interval – Class Interval, which is also known as class size, is calculated as the space between the class’s upper and lower limits.

Primary and Secondary Data – The information that the researcher personally gathers is referred to as primary data. Secondary data refers to information that is gathered by someone other than the investigator.

Variable or Variate -Any data parameter whose magnitude varies from one observation to the next is referred to as a variable or variable. Variables like age, weight, and height are a few examples.

Frequency – The number of times an observation appears in the data is its frequency.

Discrete Frequency Distribution – A frequency distribution is referred to as discrete if the exact measurements of the units are clearly displayed in the data presented.

Continuous Frequency Distribution – This type of frequency distribution is used when the precise measurements of the units are unclear and the data are presented as classes or groups.

Graphical Representation of Frequency Distribution

Histogram – Before producing a histogram, all class intervals are marked on the x-axis. Vertical rectangles are then constructed at regular intervals after that. Each rectangle’s contained area corresponds to the frequency distribution of the corresponding class interval, and each rectangle’s height is proportional to the frequency distribution of the corresponding class group.

Bar Diagram – To create a bar diagram, mark the x-axis with equal lengths for each class. Only the lengths of the bars, which are proportional to the frequency distribution of that class, are taken into account.

Pie Diagrams – Based on the relative frequency distribution, pie diagrams show data. The number of sectors in a pie diagram is equal to the number of classes in a frequency distribution. A circle is first drawn. The size and inclination of each sector depend on the relative frequency of a class.

Frequency Polygon – Ungrouped frequency distribution is represented by a frequency polygon. Plotted points have frequencies as ordinates and variate values as abscissae. The frequency polygon is then obtained by connecting these points along a straight line.

Arithmetic Mean – Calculating the arithmetic mean involves summing a group of numbers and dividing the result by the total number of groups.