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Range – General Information

December 19, 2011 Leave a comment

The range of a set of data is the difference between the highest and lowest values in the set. For example : 21, 55, 12, 84, 32, 64. Range = 84 – 12 = 72.

The range, in the sense of the difference between the highest and lowest scores, is also called the crude range. When a new scale for measurement is developed, then a potential maximum or minimum will emanate from this scale. This is called the potential (crude) range. Of course this range should not be chosen too small, in order to avoid a ceiling effect. When the measurement is obtained, the resulting smallest or greatest observation, will provide the observed (crude) range.

Median – General Information

December 19, 2011 Leave a comment
English: Comparison of mean, median and mode o...

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Median is numerical value separating the higher half of a sample from the lower half. The median of some variable x is denoted either as \tilde{x} or as \mu_{1/2}(x)\,\!.

An example for median:

To find median of 5, 9, 2, 6, 7 firstly sort by value: 2, 5, 6, 7, 9. The middlest value is median = 6. Other method to find median = [(n+1)/2]th value. (n is the number of values. median=(5+1)/2=3. 3rd value is 6.

If number of values even:

5, 9, 2, 6, 7, 1.  Sort by value: 1, 2, 5, 6, 7, 9. Median is the mean of middlest values: (5+6)/2=5.5 . Other method to find median = [(n+1)/2]th value. (n is the number of values. (6+1)/2=3.5 median is 3rd value + 0.5*|difference between 3rd and 4th value| = 5 + 0.5*(6-5) = 5.5 .

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MODE – General Information

October 15, 2011 1 comment

Some information is taken from wikipedia.

In statistics, the mode is the value that occurs most frequently in a data set or a probability distribution. Mode is the most common value obtained in a set of observations.  mean-mode approx 3(mean-median)

The mean, median and mode of a data set are collectively known as measures of central tendency as these three measures focus on where the data is centred or clustered.  To analyse data using the mean, median and mode, we need to use the most appropriate measure of central tendency.  The following points should be remembered:

  • The mean is useful for predicting future results when there are no extreme values in the data set.  However, the impact of extreme values on the mean may be important and should be considered.  E.g. The impact of a stock market crash on average investment returns.
  • The median may be more useful than the mean when there are extreme values in the data set as it is not affected by the extreme values.
  • The mode is useful when the most common item, characteristic or value of a data set is required.

13, 18, 13, 14, 13, 16, 14, 21, 13        The mode is 13, because it is repeated more often than others.

 

Inequality of arithmetic and geometric means

September 21, 2011 Leave a comment

Means – General Information

September 20, 2011 1 comment
Some information is taken from Wikipedia.

In statistics, mean has two related meanings:

for \: series \: a_1 , a_2 , a_3 \ldots \ldots a_n

Arithmetic mean (or just mean) is {\sl A.M. } = \frac{1}{n} \cdot \sum \limits_{i=1}^n a_i

Weighted arithmetic mean is {\sl W.A.M. } = \frac{ \sum \limits_{i=1}^n w_i \cdot a_i }{ \sum \limits_{i=1}^n w_i }

The weighted arithmetic mean is used, if one wants to combine average values from samples of the same population with different sample sizes. The weights wi represent the bounds of the partial sample. In other applications they represent a measure for the reliability of the influence upon the mean by respective values.

The geometric mean is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean) e.g. rates of growth.

{\sl G.M. } = \bigg( \prod \limits_{i=1}^n a_i \bigg)^{1/n}

The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed (distance per unit of time).

{\sl H.M. } = n \cdot \bigg( \sum \limits_{i=1}^n {1/a_i} \bigg)^{-1}

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