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# Averages

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 Sub Topics An average of statistical data is the value which is representative of the entire data. An average represents the entire data, its value lies somewhere in between two extremes. For this reason, statistics average is called Measure of Central Tendency. Now we discuss how we calculate statistical average: There are following methods we use to calculate statistical average - Raw method: If we have n quantities x1, x2, x3, x4,............................xn, then Mean or average of n quantities is Mean = (x1+ x2 + x3 + x4+..........................+xn)/n Direct method: if data are distributed frequency wise, then mean or average is Mean = (f1x1 + f2x2 + f3x3 + …...........fnxn)/∑fi = ∑f1xi / ∑fi Where x stands for the variable, fi stands for the frequency of xi and ∑fi stands for the total frequencies. Mean by short-cut method: The formula of short-cut method is - Mean = a + ∑fidi/∑fi = xi – a Where a is the assumed mean. Mean by step-deviation method: The formula of step-deviation method is - Mean = a + ∑fi ui/∑fi * h Where a stands for assumed mean and ui = (xi – a)/h, here h stands for class-width These are methods for calculating the average Statistics.

## Mean

The mean of any number of values is the sum of the values divided by the number of values. Suppose we have a Set of a1, a2, a3…………an then the mean Statistics of the given number set will be (a1 + a2 + a3 +………… + an) / n. this is also known as Arithmetic Mean(K).
We have several types of statistics mean or Averages for several types of data Sets.
Arithmetic mean (AM):
It is also known as standard average. Its general formula is
K = 1 / n * ∑ (i = I to n) ai
Suppose we have small number of students, some are tall while majority of them is small in height. Since the majorities are small in height therefore the mean definitely come lower than the middle value.
For example: The arithmetic mean or statistical mean of 5 values is
(5 +10+15+20+25)/5 = 75/5 = 15

Geometric Mean:
The Geometric mean is the standard average of the data sets identified by their product not by their sum. Example: Growth Rate
The general formula of geometric mean
K = (∏ (i = 1 to n) ai) 1 / n
For example the geometric mean of 6 values is
(1 * 2 * 3 * 4 * 5 * 6) 1 / 6 = 17
Hence 17 is the geometric mean of the above data set.
Harmonic Mean:
The harmonic mean is the average of data sets which are in relation to some unit. Example: Volume
The general formula of harmonic mean is
K = n * (∑ (I = 1 to n) 1/ai )-1
This was a brief description about mean statistics.

## Mid-range

We say that mid-range in Statistics is the midway Point between the minimum and the maximum in a Set of data. It is calculated as: midrange = (maximum + minimum) / 2.
For any set of numerical data, we say that the mid range in statistics of the given data is the Arithmetic. Mid-range is the mid – extreme of the set of statistical data values. It is the arithmetical Mean of the maximum and the minimum values that exist in the data set. It only deals with the outer liners of the given data and does not deal with the data inside. If the largest value and the lowest value in the data are same, it results to zero. So in such cases it cannot be used in Data Analysis. The mid-range is very commonly used to estimate the values between given two values. It is a Measure of Central Tendency. In order to reduce the chances of choosing and selecting wrong figures from the large data in the formula, we should always first try to arrange the data in ascending or descending order and then proceed for calculating midrange statistics for the data.

## Median

In mathematics or Statistics, median is defined as the value separating the data Set or the Probability distribution into two halves, lower half and upper half. The median in statistics can be determined by placing all the Numbers in increasing order and picking up the middle value. If the number of values are even than the median is defined by the arithmetic Mean of the two middle values. The median in statistics is basically used in population evaluation and probability. It is not necessary that the statistics median should be present in the data set. It may also happen that the median value may exist in the data set for one or more than one time.
Suppose we have set (p, q, r) where p < q < r then the median is ‘q’ but if we have a set (p, q, r, s) where p < q < r < s then the median is mean of ‘q’ and ‘r’. The median statistics helps in minimizing the deviation of the Arithmetic Mean. The median of the following set 3, 2, 2, 2, 1, 13 is 2.
Here the median value 2 will minimize the mean of absolute deviation (1, 0, 0, 0, 1, 11) is 2.3. Thus we can generalize median statistics as a value which minimizes the value of ‘m’. E |K - m| is a median of variable ‘K’. The value of the median ‘m’ may not be properly defined, it may also happen that the median is not even defined.

## Mode

While studying Statistics Mean, median and mode are the Basic Terms used in the statistics sampling of data. Mean is used to find the Average of the given data, Median is used to find the mid value of the given data and mode in statistics is used to find the value of the most occurring value in the given data.
Literal meaning of mode statistics is the most occurring value, which means to find the number in the given collection of data which is occurring maximum number of times. It can also be said as the data with highest frequency is called statistical mode.
To find the mode of the given data, we will first arrange the data in ascending order and then find the frequency of all the collected data. The data with the highest frequency will be called the mode. Example: find the mode of the given ages of the teachers in the school: 29, 35, 43, 28, 36,41, 35, 41, 35, 29, 30, 35
Now we first arrange the data in ascending order and get
28, 29, 29, 30, 35, 35, 35, 35, 36, 41, 41, 43
Here the frequency of 35 is maximum as 35 occurs 4 times. So we can say that 35 is the most occurring age.
So the mode is 35.