MEASURES OF CENTRAL TENDENCY II

Question 1

Briefly describe weighted mean

Answer 1

Used in situations where the relative importance of the different items is not the same. Eg in summing coursework

*See notes

Question 2

Briefly describe trimmed mean

Answer 2

A trimmed mean is calculated by discarding a certain percentage of the lowest and the highest scores and then computing the mean of the remaining scores. For example, a mean trimmed 50% is computed by discarding the lower and higher 25% of the scores and taking the mean of the remaining scores.
The median is the mean trimmed 100% and the arithmetic mean is the mean trimmed 0%. Trimmed means are often used in Olympic scoring to minimize the effects of extreme ratings possibly caused by biased judges.

Question 3

Briefly describe geometric mean and give its properties

Answer 3

This is the nth root of the product of n items or values. *See notes

Properties of the Geometric mean
1. The product of values of the series remain unchanged when the value of the GM is substituted for each individual value.
2. The sum of the deviations of the logarithms of the original deviations of the original
observations above or below the logarithm of the GM is equal. Because of this property the GM is useful in finding the average of percentages, ratios, indexes or growth rates

*See Notes

Question 4

What are the uses of Geometric mean

Answer 4

1. Finding average percent increase in sales, production, population or other economic or business series.
2. Construction of index numbers
3. I is the most suitable average when large weights have it be given to small items and small weights to large items

Question 5

What are the merits of Geometric mean

Answer 5

1. Based on all items of the series
2. It is rigidly defined
3. Useful in averaging ratios and percentages and in determining rates of increase and decrease.
4. It gives less weight to large items and more to small ones than does the arithmetic mean. Because of this GM ≤ Arithmetic mean
5. It is capable of algebraic manipulation. A combined GM can be obtained of two or more series using the formula *see notes

Question 6

What are the uses of Geometric mean

Answer 6

1. It is difficult to understand
2. It is difficult to compute and interpret
3. It cannot be computed when there are both negative and positive values in a series or one or more of the values is zero. Because of this it has a restricted application

Question 7

What is harmonic mean?

Answer 7

It is the reciprocal of the arithmetic mean of the reciprocal of the individual observations.
*See notes

Question 8

What are the uses of harmonic mean

Answer 8

It is useful in situations where the average of rates is required e.g finding average speed.
The weighted harmonic mean is used if the items do not have the same weight e.g when calculating average speed and distances covered at the different speeds differ
*See notes for examples

Question 9

What are the merits of harmonic mean

Answer 9

1. Its value is based on every item of the series
2. It lends itself to algebraic manipulation.
3. It gives better results than other averages in problems relating to time and rates.

Question 10

What are the limitations of harmonic mean

Answer 10

1. It is not easily understood
2. It is difficult to compute
3. It gives largest weight to smallest items. This makes it not useful for analysis of economic data.
4. Its value cannot be computed when there are both positive and negative items in a series or when one or more items are zero.

Because of these limitations the HM has very few practical applications except in cases where small items need to be given very high weightage.