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The following points highlight the top three kinds of mathematical average. The kinds are: 1. Mean 2. Mode 3. Median.

**Kind # 1. Mean:**

**(A) Arithmetic Mean****: **

It is most commonly used of all the averages. It is the value which we get by dividing the aggregate of various items of the same series by the total number of observations.

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**Calculation for Ungrouped Data****: **

When observations are denoted by x values showing x_{1}, x_{2}, x_{3}… x_{n}; the total number of observations is calculated by summing up the observations and dividing the sum by the total number of observations (n)

Find out the average pod length of the plant.

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**Calculation for Grouped Data****: **

When the series is discrete, each value of the variable is multiplied by their respective frequencies, sum of all values is divided by total number of frequencies. Variable x has the values like x_{1,} x_{2}, x_{3}, …, x_{n} and their frequencies are f_{1} , f_{2}, f_{3}, …, f respectively.

**Then Arithmetic Mean: **

When the series is continuous, the arithmetic mean is calculated after taking the midpoint value of class intervals.

where, x̅ = Arithmetic mean

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∑fm = Sum values of midpoint value multiplied by their frequencies

If = Sum of frequencies

m = Mid points of various class intervals.

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**Example 2: **

An observation on 32 Balsam plants shows the following data. Calculate the arithmetic mean.

The average number of flowers/plant is 6.62.

**Merits, Demerits and Uses of Arithmetic Mean: **

**Merits****: **

1. It has the simplest formula to calculate and it is easily understood.

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2. It is rigidly defined mathematical formula the same result will come on repeated calculations.

3. The calculation is based on all the observations.

4. It is least affected by sampling fluctuation.

5. The arithmetic mean balances the value on either side.

6. It is the best measure to compare two or more series.

7. Arithmetic mean is totally dependent on values not on the position.

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**Demerits****: **

1. It cannot be calculated if all the values are not known.

2. The extreme values have greater effect on mean.

3. The qualitative data cannot be measured in this way.

**Uses****: **

1. The arithmetic mean is mostly used in practical statistics.

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2. Mean helps to calculate many other estimates in statistics.

3. The arithmetic mean is most popular method of any measurement used by common people to get the average of any data.

**(b) Geometric Mean****: **

The geometric mean is defined as the n-th root of the product of n observations.

Where n = number of observations; x_{1}, x_{2}, x_{3…} x_{n} = variable values.

When n is small then the above formula can be applied but in case of large ‘n’ number the logarithms are used to find out the GM

**Example 3: **

Find out the geometric mean of the following seeds, x denotes the weight of each seed in mg.

So the geometric mean of seed weight = 5.89 mg.

This mean is based on all observations, rigidly defined, less affected by extreme values. This mean is difficult to understand, compute and interpret.

This mean is mostly helpful in averaging ratios, percentage and determining ratio of change. This mean is important in construction of index number.

**(c) Harmonic Mean****: **

When the variables are expressed in ratios or rates, the proper average to be calculated through harmonic mean. The harmonic mean is defined as the reciprocal of arithmetic mean of the reciprocal of the given values.

The harmonic mean is applicable only in restricted field such as oxygen consumption/hour, calorie requirement/hour, CO_{2} evolution/hour, flow of sap/min, etc.

Where n = Total number of observation; x_{1}, x_{2}, x_{3} are the values of variables.

**Example 4: **

In a particular experiment, 5 different sets of Hydrilla plants showed O_{2}, evolution/hour, was recorded.

So, harmonic mean of the observation is 2.17 c.c./hour. This HM determination is based on all the observations of a series. It gives more weightage to the smaller items and also not much affected by sample fluctuation.

It is not very easy to calculate and also the positive and negative, both values, cannot be computed.

**Average of Position****: **

From the data of any observation, one can find a peak in the middle with higher and lower values distributed more or Jess symmetrically towards both sides of the peak.

**Kind # ****2. Mode****: **

In a frequency distribution, ‘mode’ is defined as “the value of the variable for which the frequency is maximum”. From the definition it is clear that mode cannot be determined from a series of individual observation, always depends on the frequency of occurrence of any item.

When the concentration of data gives only one peak then the distribution is unimodal, but if the data concentrates at two or more points on a scale of values, then the series is called bimodal or multimodal.

In the Example 2, we find the maximum frequency in case of variable value 7. So the mode value of this observation is 7. This type of distribution is called unimodal distribution.

The maximum frequency (22) is observed in case of class value 27-29 (Table 9.4), the mid value of this class is 28. So, the mode value of this observation is 28.

**Example 5: **

**In another observation on 30 Balsam plants shows the following data. **

Here the mode value cannot be calculated by mere inspection, as the maximum frequency is observed in case of two values of variable 6 and 8. So to determine the modal class, the data is grouped.

**If we take 2 values together then the grouped data can be arranged in following ways: **

Here the modal class is 7-8, where mid value is 7.5, so the mode-value of this distribution is 7.5. This type of distribution is called bimodal distribution.

**Merits and Demerits of Mode:**

**Merits****: **

The mode value avoids the effects of extreme items. The value is got by mere inspection of data. All values need not to be known, it refers to a measurement which is most usual and most likely variate. The bimodal or multimodal distribution gives good indication of the heterogeneity of the population.

**Demerits****: **

This value does not need any kind of computation. It becomes difficult to comment on bimodal or multimodal distribution. This value is less dependable as all observations in a series do not have any influence on the value.

**Kind # ****3. Median****: **

The median of a distribution is defined as the value of that variable which divides the total frequency into two equal parts when the series is arranged in ascending or descending order of magnitude. So in a distribution, half of the values remain below median value and half of the values remain above the median value.

**Median Value for Ungrouped Data****: **

Median value is the value of the n+1/2 item. But this formula is applicable straightly when item number is odd. But when the item number is even, the median value is calculated by the mean value of n/2th and (n/2+1)th items

**Example 6: **

**Calculate the median number of flowers in the following observation obtained from garden plants. **

The observations are arranged in both ascending and descending order. In case of observation of 7 plants the * marked item no. should not be considered.

If we take 7 observations, then the median value will be value of 7+1/2th, i.e., 4th item, i.e., 20.

If we take 8 observations, then the median value will be the mean of 8/2th and 8/2 +1th item.

.^{.}. Mean of 4th and 5th item, i.e., mean of 20 and 21 which is 20.5.

**Median Value for Grouped Data****: **

For grouped data, the classes are arranged according to the ascending order and respective frequencies are written against them. The frequencies are then cumulated and position of the median is calculated by the same formula. The median value is the mid value of the class in which the median item value is placed.

As the total number of variables is 100, so the median value will be the value which is in between the value of 50th and 51st item value.

50th and 51st item value is in the class interval 27-29 (no. of pods).

The median value is 28 of this observation.

**Merits and Demerits of Median:**

**Merits****:**

In normal distribution, the median value is near the mean value which is more easier to calculate. This value eliminates the effect of extreme items, since they are not taken into account for its calculation, only the middle items are required to be known.

**Demerits****:**

When the distribution is irregular then the median value is not at all the true representative of the series. In case of grouped data also, the precision is not there, this value is not very useful for further analysis, as it is difficult to handle mathematically.