## Range – General Information

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

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 or as

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 .

## MODE – General Information

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.

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.

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## Inequality of arithmetic and geometric means

First of all we need to proof that It is easy, you just have to know

## Means – General Information

Some information is taken from Wikipedia.

In statistics, **mean** has two related meanings:

- the arithmetic mean (and is distinguished from the geometric mean or harmonic mean).
- the expected value of a random variable, which is also called the
*population mean*.

Arithmetic mean (or just mean) is

Weighted arithmetic mean is

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 *w*_{i} 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.

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).