Geometric mean,Arithmetic Mean,Range Quartiles

1. Business Statistics
Made by :- Ankit Singh
Class:- B.com (Hons)
Year:- 2nd Year

2. Topics
 The topics of this presentation are:-
 Geometric Mean (G.M.)
 Harmonic Mean (H.M.)
 Range
 Quartiles

3. Glance at Geometric Mean
Advantages
 It is rigidly defined.
 It is based upon all the observations.
 It is suitable for further mathematical treatment.
 It is not affected much by fluctuations of samplings.
 It gives comparatively more weight to small items.
Disadvantages
 Because of its abstract mathematical character, geometric mean is not easy to understand and to calculate for non-
mathematics person.
 If any one of the observations is negative, geometric mean becomes imaginary regardless of the magnitude of the other
items.
 It is not easy to understand.

4. Geometric Mean
Definition:- It is defined as the nth root of the product of n items.
Formula :-
Series Formula
Individual Series
n√X1*X2*X3……XN
OR Antilog(∑ log x÷ n)
Discrete Series
Antilog(∑f log x ÷ n)
Continuous Series
Antilog (∑F log m÷ n)
Where m is the mid value of each
class interval.

5. Harmonic Mean
Definition:- Harmonic Mean is the reciprocal of the arithmetic mean of the reciprocals. In other words, it is
the number of observations, divided by the sum of reciprocals of the observations.
Individual:-
Discrete and
Continuous:-

6. Glance at Harmonic Mean
Merits:-
 It is based on all observations.
 It not much affected by the fluctuation of sampling.
 It is capable of algebraic treatment.
 It is an appropriate average for averaging ratios and rates.
 It does not give much weight to the large items.
Demerits:-
 Its calculation is difficult.
 It gives high weight-age to the small items.
 It cannot be calculated if any one of the items is zero.
 It is usually a value which does not exist in the given data.

7. Range
Definition:-The Range is the difference between the lowest and highest values of a series.
Formula:- H-L
Coefficient:- H-L ÷H+L

8. Glance at Range
Merits or Uses
 It is easiest to calculate and simplest to understand even for a beginner.
 It gives us the total picture of the problem even with a single glance.
 It is used to check the quality of a product for quality control.
 Meteorological Department also makes forecasts about the weather by keeping range of temp.
Disadvantages:-
 Range is not based on all the terms. Only extreme items reflect its size. Hence range cannot be completely
representative of the data as all other middle values are ignored.
 Range does not change even the least even if all other, in between, terms and variables are changed.
 It does not tell us anything about the variability of other data.
 For open-end intervals, range is indeterminate because lower and appear limits of first and last interval are not given.

9. Quartiles
 Definition:- Quartiles refers to the each of four equal groups into which a population can be divided
according to the distribution of values of a particular variable.
Elements of the Quartiles:- There are four elements of
the quartiles which depicts the four arena of a population or an object like for e.g.
circle, rectangle etc.
 They are listed below:-
Q1:- First quarter of an object. ─ ( Nth /4 of an observation)
Q2:- Second quarter of an object . ─ (2nth /4 of an observation)
Q3:- Third quarter of an object . ─ (3rd /4 of an observation)
Q4:- Not a quarter but an object . ─ ( nth of an observation or a object)

10. How to Calculate
 First of all in a given series, calculate nth terms.
In case of individual series, the total no. of the observation is the nth terms.
In case of discrete series and continuous series ,the last cumulative frequency is the nth terms.
In case of inclusive and exclusive series , the last cumulative frequency is the nth term.
In case of more than series , the first frequency is the nth term while in less than series the last frequency is
the nth term but it is preferable to convert it into the continuous series and convert the frequency into cf.
 Second step is to find Quartiles whether Q1,Q2,Q3 depending on the question asked.
 Put it into the formula.
Formula:-𝑙 +
𝑗𝑛
4
𝑓
−
𝑐𝑓
𝑓
∗ 𝑖

11. Quartiles and its related concept
 The Quartiles and its related concept are as follows:-
Quartiles Concept Formula
Quartile Deviation Q3-Q1 ÷ 2
Coefficient of Quartile
Deviation
Q3-Q1÷ Q3+Q1
Inter Quartile Range Q3-Q1

12. Glance at Quartiles
Merits:-
 They can be directly determined in case of an open end series without locating the lower limit of the lowest
class, and the upper limit of the highest class.
 They are useful in the computation of the measures of dispersion and skewness.
 They give an idea about the character of a frequency distribution i.e. whether a series is symmetric, or
asymmetric can be known by measuring their distance from the Median.
 They are not affected very much by the extreme values of a series.
Demerits:-
 These averages are not easily understood by a common man.
 They are not based on all the observations of a series.
 They need the rearrangement of series in the ascending order if given otherwise.
 They do not study the entire data.

13. Question on Quartiles Deviation
 Q) The following are the marks obtained by the student in Accountancy.
Calculate Q1, Quartile Deviation, Coefficient of QD, Inter Quartile Range.
Marks Frequency of student
20-30 3
30-40 61
40-50 132
50-60 154
60-70 140
70-80 51
80-90 3

14. Answer
Answer:-
Nth term:- 544
 Q1:- 45.45
Q3:- 64.14
Inter Quartile Range:- 18.69
Quartile Deviation:-9.345
Coefficient of QD:- 0.17