## Short Questions and Answers about Measure of Central Tendency

This page contains short questions and answers about *measure of central tendency * which includes averages and types of averages such as
Mean, Median, Mode, Weighted Mean, Relationship between averages etc.

**Measure of Central Tendency : **The tendency of the observation to cluster in the central part of the data set is called Central tendency and the summary
value is called measure of central tendency . The measures of central tendency are generally known as Averages.
The most common central tendencies are the arithmetic mean (simple average or mean), the median, and the mode.
Measure of central tendency also called measure of location.

**Average ** is a single value used to represent the distribution. Most commonly used averages are Mean, Median and Mode.

**Mean: ** The arithmetic mean or simply mean is the statistician's term also knows as the average, which is obtained by dividing the sum of all the
observations by total number of observations (summed).

**MEDIAN:** The median is the middle value of the series when the variable values are placed in order of magnitude (in ascending or descending order).

**MODE:** The mode is defined as that value which occurs most frequently in a set of data i.e. it indicates the most frequent (common) results.

Note that the median indicates the middle position while the mode provides information about the most frequent value in the data. Both median and mode are different methods of calculating the average value of data and they have their own advantages and disadvantages. They are used by the statisticians according to their requirement.

- Arrange the values in increasing or decreasing order.
- Count the number of values.
- If the number of observation in data set is odd then Median is $\frac{(n+1)}{2}$th value, and if the number of observation in data set is even then Median is the average of $\frac{n}{2}$th and $[(\frac{n}{2})+1]$ th observations.

Note that the median is the value halfway through the ordered data set (arranged in ascending or descending order), below and above which there lies an equal number of data values. The median is the second quantile.

To find the grouped data, follow these steps:In Case of frequency distribution (group data) first we find the median group (median class) by
$\frac{n}{2}$. Then use the following formula to find the median of data. $Median = l + \frac{h}{f}(\frac{n}{2}-C)$

where

$l$ = lower class limit
(lower class boundary) of median class,

$h$ = class interval,

$f$ = frequency of median group

and

$C$ = Cumulative frequency of the class above the median
group.

The **geometric mean ** is a measure of central tendency, it uses multiplication rather than addition to summarize data values. The geometric mean is a
useful measure of summary when we expect that changes in the data occur in percentages. For example adjustments in salary are often a percentage amount.
Geometric means are often useful for highly skewed data. Don't use a geometric mean, though, if you have any negative
or zero values in your data.

The **Geometric Mean ** is used primarily to average data for which the ratio of successive terms remain approximately constant.
This occurs with data as rate of change, ratios, economic index numbers, and population sizes over successive time periods etc.

**Harmonic Mean ** is most frequently used in average speeds of various distances covered, where the distances remain constant. It is also used in finding
the average cost of commodity, such as mutual funds, when several different purchases are made by investing the same amount of money each time.

Relation between arithmetic mean, geometric mean and harmonic mean is given below:

*Arithmetic Mean > Geometric Mean >Harmonic Mean * i.e. for a data, arithmetic mean is greater than geometric mean and harmonic mean, and geometric
mean is greater than harmonic mean.

**Weighted Mean: ** The value of each observation is multiplied by the number of times it occurs (or by the weight of the observation). The sum of these
products is divided by the total number of observations to give the weighted mean. Note that, the multipliers or a set of numbers which express more or less
adequately the relative importance of various observations in a data set are technically called the weights. The formula of weighted mean is as follows.

**Weighted Mean =** $\frac{\sum x_i w_i}{\sum w_i}; \quad \mbox{where } i=1,2,\cdots,n$.

**Mid Range:** If there are *n* observations with $X_0$ and $X_m$ as their smallest and largest observations respectively, then their mid-range is
defined as *Mid range = * $\frac{X_0+X_m}{2}$ i.e. Mid range is the arithmetic mean of the smallest and largest value in data set. It is obvious that
if we add the smallest value with the largest, and divide by 2, we will get a value which is more or less in the middle of the data-set.

Central Tendency means the tendency of the data to gather around some central value and the value around which all the observations tend to gather is called measure of central tendency. Measure of Central Tendency or central tendency are generally known as Averages. The most common types of averages are:

- Arithmetic Mean (AM)
- Geometric Mean (GM)
- Harmonic Mean (HM)
- Median
- Mode
- Weighted Mean (WM)