Statistics-Chapter 3 (Measures of Location)

Q.1) What is a statistical average and objects of its measurement?

Answer:
Statistical average:
Statistical average is the value that represents the whole data of a series and thus, through the average, the large values running into thousands and lakes can be represented by a single value.
Dr. Bowley defines an average in the following term:
“An average is purely mathematical concepts such as the average length of the life in a varied population which does not correspond to any particular group but is only short way of expressing an arithmetical result”
Objects of statistical average:
· The most important object of measuring location or average is to find out a value that can represents the whole data.
· One of the objects of measuring average is very much useful for comparative study of different distributions.
· Averages are very useful for computing various other statistical measures such as dispersion, skewness, kurtosis and various other basic characteristics of a mass of data.

Q.2) What do you understand by weighted mean? Explain the difference between a simple mean and weighted mean.

Answer:
Weighted mean:
In calculating the arithmetic average, we give equal importance to all the items of the series but it is not necessary that all the constituents of the total number of items have the same relative degree of importance. Then the relative importance of the various items is indicated by assigning “weights” to have various items. The average thus computed by making use of the weights is called “weighted arithmetic average”.

Q.3) What are the difference between Mathematical averages and averages of position.

Answer:
Mathematical averages:
Arithmetic average:
One of the most common used measures of location is the arithmetic average or mean. Arithmetic mean is mid value of the item in data. Mean is called as the central value of a distribution.
Geometric mean:
If x₁ x₂,…,x_n are the n values of a data set, thus mathematically geometric mean is given by ;

It means that the G.M.of a set of n values is the nth positive root of their product. The calculation took may be simplified by taking common logarithms and anti-logarithm of formula
Harmonic mean
Harmonic mean of a series of values is the reciprocal of the arithmetic mean of the reciprocal of the values of items.

Averages of position:
The mathematical averages, (arithmetic mean, geometric mean and harmonic mean) cannot be calculated if any value is missing, negative and zero. Extreme items of the series badly affected the value of A.M, G.M and H.M to a very large extent. For example a group of 7 students gives the intelligence test, obtained the following marks (x):
x = -1, 0,3,4,6,7,20
4 is the representative figure but A.M = 5.6, G.M and H.M is not
Defined because of -1 and 0. So all the three averages fail to give a representative figure. Hence we need other type of averages that is averages of position such as median and mode.
Median:
Median is the value of the middle item of a series when it is arranged in ascending or descending order of magnitude. Median divides the series in two equal parts. Median is located in such a way that in one part; the items are less than or equal to median and in other part the items are more than or equal to median.
Quartiles:
A Quartile is the value of item divided into four equal parts as the median is the value of middle item of a series of data. There are two quartiles
Q, is the first quartile and Q₃ is the third quartile.
i.e.
The Q, and Q₃ are also called the lower and upper quartiles
respectively. The second quartile (Q₂) is the median.
Deciles
They divide the data into ten equal parts, denoted by D₁,D₆. The fifth decile D₅ coincides with the median value. A method similar to that used for calculating the quartiles.
Percentiles
They divide the data into hundred equal parts. The first, second, ninety ninth percentiles are denoted by Pl₅ P₂, P₉₉ respectively. The P₅ coincides with the median.
Mode
Mode is the most frequent value in a data set. In other words the maximum point of the frequency distribution.

Q.4) What are the requisites of a representative average?

Answer:
Requisites of representative average:
Following are the essential qualities of an ideal average or measures of location.
· It should be rigidly defined and left to guessing of the observer.
· A good average should be based on all the observations of the series.
· Otherwise it cannot he a representative value a good average must possess the readability to further algebraic interpretations.
· A good average should be easily understandable.
· A representative average must not be affected by the fluctuations of sampling.
· It should not be affected much by extreme observations.
. It should be capable of being easily and rapidly calculated.

Q.5) Define arithmetic mean and its four properties. Also prove that

Σ (x – x)₂ < Σ (x – a )₂

Solution:

Q.6) Define geometric mean, mean and harmonic mean.Prove that A.M > G.M > H.M.

Answer:
Geometric Mean
The geometric mean is the nth root when you multiply n numbers together. It is not the same as the arithmetic mean, or average, that we know. For the arithmetic mean, we add our numbers together and divide by how many numbers we have. The geometric mean uses multiplication and roots. For example, for the product of two numbers, we would take the square root. For the product of three numbers, we take the third root.

Harmonic mean:

The mean of a set of positive variables. Calculated by dividing the number of observations by the reciprocal of each number in the series.

Mean:

The mean is the average of all numbers and is sometimes called the arithmetic mean. To calculate mean, add together all of the numbers in a set and then divide the sum by the total count of numbers. For example, in a data center rack, five servers consume 100 watts, 98 watts, 105 watts, 90 watts and 102 watts of power, respectively. The mean power use of that rack is calculated as (100 + 98 + 105 + 90 + 102 W)/5 servers.

PROVE OF A.M>G.M>H.M

A.M=(a+b) /2 = (16+25) / 2 =41/2 = 20.5

G.M=(a.b) =(16×25) = 4×5 = 20

H.M=2(ab) / (a+b) =2 (16 x 25) / (16 + 25) =19.5

So hence it is proved that :

A.M > G.M > H.M= 20.5> 20 > 19.5

Q.7) Show that the geometric mean lies between arithmetic mean and harmonic mean of the two values 16 and 25.

Solution:
A.M= (a+b) /2 = (16+25) / 2 =41/2 = 20.5
G.M= (a.b) =(16×25) = 4×5 = 20
H.M=2(ab) / (a+b) =2 (16 x 25) / (16 + 25) =19.5
A.M > G.M > H.M
20.5> 20 > 19.5

Q.8) Describe the relative advantages and disadvantages of the mean, geometric mean and harmonic mean.

Answer:
Arithmetic mean
Advantages
· It is very easy to calculate and very useful measure of location.
· It is easy to understand.
· Mean helps in comparison the items even when their number is very large.
· Arithmetic average possess the quality of being put to further comparison.
· It is important to descriptive statistics.
· It is more useful in statistical inference than any other measures of locations, because of its mathematical properties.
· It is mathematically defined value that is amenable to mathematical treatments.
· Of all averages, arithmetic mean is affected least by variations of sampling. This property is explained by saying that arithmetic mean is a stable average.
Disadvantages
· Extreme items of the series affect the value of mean to a very large extent. For example mean of the 4, 5,6,7,8,9,10 is 7, but if we include a value 103 to this set. The mean will become 19.
· It is not suitable for qualitative data.
· It often gives misleading conclusions.
· It’s difficult to locate mean by more inspection.

Geometric mean
Advantages
· It is based on the whole data of a series.
·As compared to other averages the marginal value have less effect on geometric mean.
· It is more convenient, for ratios, rates and percentages
· It is suitable for further mathematical treatment.
· It is not affected much by variations of sampling.
Disadvantages
· It is difficult to calculate geometric mean. If the value of a variable is zero or negative then it is not possible to calculate geometric mean.
· It gives more importance to smaller values.
· The geometric mean may not correspond to any actual value of the data set.

Harmonic mean
Advantages
· Every item affects the value of harmonic mean
· Like Arithmetic mean, this average can also be treated mathematically.
· It strikes a balance by giving more importance to small values and less importance to big values.
· Harmonic mean is more suitable where there is much skewness or dissimilarity in the series.
Disadvantages
· Harmonic mean cannot be obtained if any one of observation is zero.
· Harmonic mean is rarely use practically, because this average is difficult to arithmetic mean.
· It is not easy to understand and difficult to compute.
· It cannot be located by inspection nor can it be located graphically.

Read more: Statistics Chapter 2 (Collection and presentation of Data) | Class 11 Notes

Q.9) State the empirical relation between mean, median and mode.

Answer:
Empirical relation between mean median and mode
In case of a symmetrical distribution mean, median and mode coincide i.e.
Mean = Median = Mode
In moderately skewed distributions,
Mode = mean – 3(mean -median)
Mode = 3 median – 2 mean
Mean – mode = 3 (mean – median)
Mode = 3 median – 2 mean

Q.10) Find A.M, G.M and H.M from the following data if possible, if not possible give reason. -1,2,3,100,89,31,0,49,50,70.

Answer:
The arithmetic mean cannot be calculated, because of extreme items of the series badly affected the value of A.M.
The geometric mean cannot be calculated, because of -1 & 0.
The harmonic mean cannot be calculated ,because of 0

Read more: Statistics Class 11 Notes Cha 1 (Introduction To Statistics)