This GSEB Class 11 Commerce Statistics Notes Chapter 3 Measures of Central Tendency covers all the important topics and concepts as mentioned in the chapter.
Measures of Central Tendency Class 11 GSEB Notes
Meaning of Measure of Central Tendency:
Central Tendency:
In classified data the values of the variable are concentrated around a certain central value. This characteristic of data is called central tendency.
Measure of Central Tendency:
The central value around which the values of the variable are concentrated is called as measure of central tendency.
Average:
A measure, representative for the whole set of data and representing an essence of the main characteristics of the data is called average. Its’ measure being in the centre of the data, is known as measure of central tendency.
Measures of Central Tendency:
- Arithmetic Mean or Mean
- Geometric Mean
- Median, Quartiles, Deciles, Percentiles
- Mode
Characteristics of an ideal of Average:
- Its definition should be clear and precise.
- It should be easy to understand and calculate.
- It should be based on all the observations of the data.
- Suitable for further algebraic operations
- A stable measure
- Less affected by too large or too small observations
- Useful for data analysis
- Real measure k Arithmetic Mean or Mean:
Mean:
The value obtained by dividing the sum of all observations of the given data by the total number of observations is called mean of the data. It is denoted as x̄.
x̄ = \(\frac{\Sigma x}{n}\) Where, Σx = Sum of observations
n = Number of observations
Combined Mean:
If the number of observations and their means are given for two or more groups, the mean obtained by combining all the groups is called combined mean. It is denoted as xc.
Weighted Mean:
The mean calculated by the observations of the given data with the consideration of their relative weights is called weighted mean. It is denoted as x̄w.
Geometric Mean:
If ‘n’ observations of given data are positive and non-zero, say x1, x2, x3, …, xn then nth root of the product of n observations is called the geometric mean of the data and it is denoted by G.
G = \(\sqrt[n]{x_{1} \cdot x_{2} \cdot x_{3} \ldots \ldots x_{n}}\)
[Note: If x̄ = Arithmetic mean of n positive observations and G = Geometric mean of n positive numbers, then inequality x̄ ≥ G is satisfied.]
Measures of Positional Average:
Median:
The value of the observation located exactly in the middle in the sequence of observations arranged in ascending or descending order of their magnitudes is called the median of the data. It is denoted by the symbol M.
M = Value of \(\frac{(n+1)}{2}\)th observation
Here, n = number of observations
Quartiles:
The quartiles are the values of observations which divide the sequence of observations of data arranged in ascending or descending order of their magnitudes into four equal parts. There are three quartiles and they are denoted by symbols Q1 Q2 and Q3.
Qj = Value of \(\frac{j(n+1)}{4}\)th observation
Here, j = 1, 2, 3
Its clear that, Q2 = Second quartile = Median; Q1 ≤ Q2 ≤ Q3
Deciles:
The deciles are the values of observations that divide the observations of the data arranged in ascending or descending order of their magnitudes into ten equal parts. There are nine deciles and they are denoted by symbols D1, D2, D3, …, D9.
Dj = Value of \(\frac{j(n+1)}{4}\)th observation
Here, j = 1, 2, 3, …, 9 It is clear that, D5 = Q2 = M
D1 ≤ D2 ≤ D3 ≤ … ≤ D8 ≤ D9
Percentiles:
The percentiles are the values of the observations which divide the observations of the data arranged in ascending or descending order of their magnitudes into 100 equal parts. There are 99 percentiles. They are denoted by symbols P1, P2, P3, …, P99.
Pj = Value of \(\frac{j(n+1)}{4}\) th observation
Here, j = 1, 2, 3, …, 99
It is clear that, P25 = Q1; P50 = D5 = Q2 = M;
P75 = Q3’ P10 = D1 P20 = D2
Thus, P10 × j = Dj, J = 1, 2, …, 9
Pi ≤ P2 ≤ P3 ≤ … ≤ P98 ≤ P99
Mode:
Mode: The value of the observation which is repeated maximum number of times in the given data is called the mode of the data. It is denoted by M0.
Relation among the measures of Average:
For the frequency distribution in which the observations of the data are not evenly distributed around average Karl Pearson established the following relation:
3 (Median – Mode) = 2 (Mean – Mode)
⇒ M0 = 3M – 2x
Where, M0 = Mode; M – Median; x = Mean
Some Algebric results for measures of Central Tendency:
- If the values of all observations of the data are equal, then the values of all measures of central tendency are equal.
For example, if the marks of statistics of 5 students of std. 11 are 70, 70, 70, 70, 70, then x = M = M0 – G = xw = 70. - For the data evenly distributed from average, Mean = Median = Mode.
- The values of Mean, Median and Mode are changed by the change of origin and scale.
For example, if multiplying the variable x by non-zero b and adding a to it, then new variable we get y = bx + a. Therefore, ȳ = bx̄ + a; Median of y =b (M) + a; Mode of y = b(M0) + a.
Comparative study of Mean, Median and Mode:
- The very popular measure of average is mean which is useful for study in advanced statistical method.
- For qualitative data median is useful. When the values of the variable are not evenly distributed, it is useful for studying social problems, business activities and agriculture .related problems.
- In business and commerce mode is more useful.
The selection of average is based on
- Nature of the data,
- Nature of variable involved,
- The purpose of study,
- The type of classification used and
- The need of statistical analysis.
Important Formulae:
1. Mean:
Ungrouped Data:
Direct method (By definition)
x̄ = \(\frac{\Sigma x}{n}\)
Where, x = Observation
Σx = Sum of observations
n = Total number of observations
Short Cut Method:
x̄ = A + \(\frac{\Sigma d}{n}\)
Where, A = Assumed mean
d = Deviation from Assumed mean
Σd = Sum of deviations
n = Total number of observations
Grouped data:
Discrete Frequency distribution:
Direct method:
Discrete x̄ = \(\frac{\Sigma f x}{n}\)
Where, x = Observation
f = Frequency of observation
n = Total frequency = Σf
Short cut method:
Discrete x̄ = A + \(\frac{\Sigma f d}{n}\)
Where, A = Assumed mean
d = (x – A)
f = Frequency of observation
n = Total frequency = Σf
Continuous Frequency distribution:
Direct Method:
x̄ = \(\frac{\Sigma f x}{n}\)
Where, x = Mid value of class
f = Frequency of class
n = Total frequency = Σf
Short Cut Method:
x̄ = A + \(\frac{\Sigma f d}{n}\) × c
Where, d = \(\frac{x-\mathrm{A}}{c}\)
x = Mid value of class
A = Assumed mid value
c = Class length
n = Total frequency = Σf
2. Combined Mean
x̄ = \(\frac{n_{1} \bar{x}_{1}+n_{2} \bar{x}_{2}+n_{3} \bar{x}_{3}+\ldots+n_{k} \bar{x}_{k}}{n_{1}+n_{2}+n_{3}+\ldots+n_{k}}\)
Where, xÌ„1, xÌ„2, …, xÌ„k are means of k groups respectively
n1, n2, …………nk are number of observations of k groups respectively
xc = Combined mean of k groups
3. Weighted Mean
x̄w = \(\frac{x_{1} w_{1}+x_{2} w_{2}+x_{3} w_{3}+\ldots+x_{k} w_{k}}{w_{1}+w_{2}+w_{3}+\ldots+w_{k}}=\frac{\Sigma x w}{\Sigma w}\)
Where, x̄w = Weighted mean; Σm = Total weight
x1, x2, …, xk are k observations respectively;
x = Observation
w1, w2, …….. wk are weights of k observations respectively;
w = Weight of observations
Some algebraic results about Mean:
1. The sum of the deviations of the observations of the data from their mean is always zero, i.e., Σ (x – xÌ„) = 0.
For example:
x: 5, 7, 9, 11, 13 x̄ = \(\frac{\Sigma x}{n}=\frac{45}{5}\) = 9
2 (x – xÌ„) = (5 – 9) + (7 – 9) + (9 – 9) + (11 – 9) + (13 – 9) = (-4) + (-2) + 0 + 2 + 4 = – 6 + 6 = 0
2. If each observation of the data is multiplied by a non-zero constant ‘b’ and a non-zero constant ‘a’ is added to it, the mean of new observations becomes bx + a.
For example:
In the above example xÌ„ = 9. If each observation is multiplied by 3 and 2 is added to it, then mean of new observations = 3(9)4 – 2 = 29.
In the same way, if a constant ‘a’ is added to each observation and the result is divided by a non-zero constant ‘c\ then the mean of new observations becomes = \(\frac{\bar{x}+a}{c}\)
For example:
In the above example x = 9. If 3 is added to each observation and then the result is divided by 4, then the mean of new observations = \(\frac{9+3}{4}=\frac{12}{4}\) = 3.
3. x̄ = \(\frac{\Sigma x}{n}\) So out of three values x̄, Σx and n, if the values of any two are given, the value of third can be obtained.
For example:
In the above example x = 9 and if n = 5 is given, then Σx = n.x = 5 x 9 = 45
In the same way, x = 9 and Σx = 45 is given, then n = \(\frac{\Sigma x}{\bar{x}}=\frac{45}{9}\) = 5
4. Geometric Mean
Ungrouped data:
G = \(\sqrt[n]{x_{1} \cdot x_{2} \cdot x_{3} \cdots x_{n}}\)
Note: The relation between mean x and Geometric mean G is x̄ ≥ G.
5. Median
Ungrouped data:
M – Value of \(\left(\frac{n+1}{2}\right)\)th observation
Where, n = Number of observations
Discrete frequency distribution:
Grouped data:
M = Value of \(\left(\frac{n+1}{2}\right)\)th observation
where, n = Total frequency = Σf
Continuous frequency distribution:
Class for median = Class in which \(\frac{n}{2}\)th observation lies
M = L + \(\frac{\frac{n}{2}-c f}{f}\) × c
Where, L = Lower boundary point of median class n = Total frequency = Σf
cf = Cumulative frequency of class preceding the median class
f = Frequency of median class
c = Class length of median class
6. Quartiles
Ungrouped data:
Qj = Value of j\(\left(\frac{n+1}{4}\right)\) th observation; Where j = 1, 2, 3 and n = Total number of observations
Grouped data:
Discrete frequency distribution:
Q1 = Value of \(\left(\frac{n+1}{4}\right)\)th observation
Q2 = Value of 2\(\left(\frac{n+1}{4}\right)\)th = \(\left(\frac{n+1}{2}\right)\)th observation
Q3 = Value of 3\(\left(\frac{n+1}{4}\right)\)th observation Where, n = Total frequency = Σf
Continuous frequency distribution:
First Quartile:
Q1 class = Class in which j\(\left(\frac{n}{4}\right)\)th observation lies
Q1 = L + \(\frac{\frac{n}{4}-c f}{f}\) × c
Third Quartile:
Q3 class = Class in which 3\(\left(\frac{n}{4}\right)\)th observation lies
Q3 = L + \(\frac{3\left(\frac{n}{4}\right)-c f}{f}\) × c
Note:
Get, either the class for Q1 or Q3 or the values of Q1 or Q3 by referring to the order of observation in the column of cumulative frequency cf.
7. Deciles
Ungrouped data:
Dj = Value of j\(\left(\frac{n+1}{10}\right)\)th observation;
Where j = 1, 2, 3, …, 9 and n = Total number of observations
Grouped data:
Discrete frequency distribution:
Dj = Value of j\(\left(\frac{n+1}{10}\right)\)th observation; Where j = 1, 2, 3, …, 9 and n = Total frequency = Σf
Continuous frequency distribution:
Class for Dj = Class in which j\(\left(\frac{n}{10}\right)\)th observation lies
Dj = L + \(\frac{j\left(\frac{n}{10}\right)-c f}{f}\) × c Where j = 1, 2, 3 9
Note: Get the values of Dj or class for Dj by referring to the order of observations in the column of cumulative frequency cf.
8. Percentile
Ungrouped data:
Pj = Value of j\(\left(\frac{n+1}{100}\right)\)th observation: Where, j = 1.2, 3, …,99 and n = Total number of observations
Grouped data:
Discrete frequency distribution:
= Value of j \(\left(\frac{n+1}{100}\right)\)th observation: Where, j = 1. 2, 3 99 and n = Total frequency =Σf
Continuous frequency distribution:
Class interval for Pj = Class in which j\(\left(\frac{n+1}{100}\right)\)th observation lies
Pj = L + \(\frac{j\left(\frac{n}{100}\right)-c f}{f}\) × c; Where, j= 1. 2, 3, ………….. 99
Note: Get the values of P or class for P by referring to the order of observations In the column of cumulative frequency cf.
9. Mode
Ungrouped data:
M0 = The observation which is repeated for maximum number of times
Grouped data:
Discrete frequency distribution: M0 = Value of the observation with maximum frequency
Continuous frequency distribution:
Modal class = A class with maximum frequency
M0 = L + \(\frac{f_{m}-f_{1}}{2 f_{m}-f_{1}-f_{2}}\) × c
Where, L = Lower boundary point of the model class;
f1 = Frequency of the class preceding the model class;
Jm = Frequency of the model class;
f2 = Frequency of the class succeeding the model class; c = length of the model class
Emperical Formula:
(For bimodal frequency distribution or frequency distribution of unequal class length)
M0 = 3M – 2xÌ„; Where xÌ„ = Mean, M = Median, M0 = Mode
Graphical Method for Mode:
- It is used only for uni-modal frequency distribution.
- It is used for the continuous frequency distribution with equal class length and unequal class length.
- To find mode by graph Histogram is used.
- To draw histogram, convert inclusive type continuous frequency distribution into exclusive type of continuous frequency distribution by class boundary points.
- If the frequency distribution is of unequal class length, find the proportionate frequency of each class by using the following formula: frequency of a class
Proportionate frequency = \(\frac{\text { frequency of a class }}{\text { class length of a class }}\) × smallest class length