Measures of Central Tendency(Part-1)| Giri Banarjee Statistics Solution| Problmes on Mean

Q1.What do you mean by central tendency of frequency distribution?What are its measures?

Ans: Central tendency gives us idea about the concentration of the values in the central part of the distribution.

1.Arithmetic mean

2.Geometric Mean

3.Median

4.Mode

5.Harmonic Mean 

Q2. State the principle characteristics of a good average.

Ans: 

a. It should be rigidly defined.

b.It should be readily comprehensible and easy to calculate.

c.It should be based on all the observation.

d. It should be should be suitable for further mathematical treatment.

 

Q3. Prove that arithmetic mean affected by change of scale and origin both.

Ans: Let, $\small y=a+bx$

So, $\small \frac{y-a}{b}=x$, here $\small a,b$ is the location(change of origin) parameter, scale parameter.

So,$\small \sum_{i=1}^{n}y_{i}=na+b\sum_{i=1}^{n}y_{i}$

or, $\small \bar y= a+b\bar x$

or, $\small \frac{y-a}{b}=x$

or, $\small \frac{\bar y-a}{b}=\bar x$

Arithmetic mean of $\small x$(variable got after changing the origin and the scale of y) is depend on the location and scale parameter.

 

Q4. If $\small \bar x_{1}$, $\small \bar x_{2}$ are the means of two sets, then show that the combined mean of the observations lie between $\small \bar x_{1}$ and $\small \bar x_{2}$.

Ans: Without loose of generality we may assume $\small \bar x_{1} \leq \bar x_{2}$.

or,$\small n_{1}\bar x_{1}\leq n_{1}\bar x_{2}$

 

or,$\small \small n_{1}\bar x_{1}+ n_{2}\bar x_{2}\leq n_{1}\bar x_{2}+n_{2}\bar x_{2}$

[Dividing both side by $\small (n_{1}+n_{2})$]

 

or,$\small \frac {n_{1}\bar x_{1}+ n_{2}\bar x_{2}}{n_{1}+n_{2}}$ $\small \leq$ $\small \frac { n_{1}\bar x_{2}+n_{2}\bar x_{2}} {(n_{1}+n_{2})}$

 

or, $\small \bar x \leq \bar x_{2}$

Similarly we can say

or, $\small \bar x_{1} \leq \bar x$

 Q5.Give some examples where median would be the most appropriate measures of average. 

Ans: A common and popular example of median is "Median salary" for a country or city.When the two average income of a country/city is discussed then median is the best measures of central tendency,because it represents the middle of a group getting that amount of salary.

Q6. Examine whether the different averages are affected when all the variable values are increased or decreased by the same amount or same proportion. Show the results.

Ans: Let, $\small a$ be the constant, and all the variables increased or decreased by the same amount or same proportion(say, $\small a$).

Let, $\small y_{i}= a \pm  x_{i}$

or,  $\small \bar y= a\pm \bar x$

Let, $\small u_{i}= a x_{i}$

or,   $\small \bar u= a \bar x$

Let, $\small v_{i}= \frac{ x_{i}}{a}$

or,   $\small \bar v =\frac {\bar x}{a}$ 

Prove for $\small G.M, H.M$ also.

Q7.Suppose each value of a variable lies between p and q., both values inclusive. Show that $\small p\leq \bar x \leq q$, where $\small \bar x$ denotes the mean of $\small x$.

Ans: According to the question,each $\small x_{i}$ lies between p and q, for all $\small i=1(1)n$.

So, $\small np\leq \sum_{i=1}^{n}x_{i}\leq nq$

or,  $\small \frac{np}{n}\leq \frac{ \sum_{i=1}^{n}x_{i}}{n}\leq \frac{nq}{n}$

or,  $\small p\leq \bar x \leq q$

 Part-2

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