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## Mean , Median , Mode

**MEAN**

**Calculation of Ungrouped Data :**

The mean of distribution is commonly understood as the arithmetic average. It is computed by dividing sum of all the scores by the number of measures. We denote the mean by ‘M’, the first letter of the word Mean. The formula is,

**Calculation of Mean from Grouped Data :**

When the scores are more in the given data, the above method is not useful then the given data is to be grouped into frequency distribution and then mean is calculated by different methods namely, the Short Method and Long Method.

**Advantages of the Mean :**

- Reflects the values of all the scores in the distribution.
- Has many desirable statistical properties
- Is the most reliable for drawing inferences, and the easiest to use in
- Advanced statistical techniques.

**Disadvantages of the Mean :**

- Usually not an actual score in the distribution.
- Not appropriate for use with ordinal data.
- Can be misleading when used to describe a skewed distribution.
- Can be strongly affected by even just one very extreme score (i.e., an outlier).

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**MEDIAN **

Median is the middle most point of a distribution or it is a midpoint in the given series. In other words, in the distribution the half of the values lies below and above the midpoint. It is measure of position rather than of magnitude. It is the 50th percentile point in the given distribution.

** Calculation of Ungrouped Data :**

The scores have to be arranged either in ascending or descending order. Find out the middle score by dividing the total scores by two. Count from one end of the series to find out the below and above which N/2 of scores life. This point is the median.

**Example :** 6, 7, 8, 9, 10, 11, 12. Here there are seven values. The middle score in the series is 9. Because, below and above 9, there are three values. In computation two situations arise. When N is odd and when N is even.

- When the number is odd : If we have an odd score and if no scores are repeated, the median is the middle value.
- When the number is even : If we have the even scores, the median will become the average of the two middle values.

**Calculation of Grouped Data :**

When the data is classified into frequency distribution, the median can be computed by using the formula given below.

**Advantages of the Median :**

- Can be used with either ordinal or interval/ratio data.
- Can be used even if there are open – ended categories or undeterminable scores on either side of the distribution.
- Provides a good representation of a typical score in a skewed distribution; is not unduly affected by extreme scores.
- Minimizes the sum of the absolute deviations (i.e., the sum of score distances from the median – – ignoring sign – – is less than it would be from any other location in the distribution).

** Disadvantages of the Median :**

- May not be an actual score in the distribution (e.g., if there are an even number of scores, or tied scores in the middle of the distribution).
- Does not reflect the values of all the scores in the distribution (e.g., an extreme score can be moved even further out without affecting the median).
- Compared to the mean, it is less reliable for drawing inferences about a population from a sample, and harder to use with advanced statistics.

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**MODE **

The mode is defined as the element that appears most frequently in a given set of elements. Using the definition of frequency given above, mode can also be defined as the element with the largest frequency in a given data set. For a given data set, there can be more than one mode. As long as those elements all have the same frequency and that frequency is the highest, they are all the modal elements of the data set.

**Example : **

Find the mode of the following data set : 48 44 48 45 42 49 48

**Solution : **

The mode is 48 since it occurs most often.

**Steps : **

- It is possible for a set of data values to have more than one mode.
- If there are two data values that occur most frequently, we say that the set of data values is bimodal.
- If there is no data value or data values that occur most frequently, we say that the set of data values has no mode.

** Calculation of Grouped Data :**

As we saw in the selection on data, grouped data is divided into class interval. We have defined mode as the element which has the highest frequency and the mode itself, which calculate from the modal class interval using the formula below.

**Advantages of the Mode :**

- Easy to find.
- Can be used with any scale of measurement.
- The only measure that can be used with a nominal scale.
- Corresponds to actual score in the distribution.

**Disadvantages of the Mode : **(the following apply when the mode is used with ordinal or Interval/ratio data) :

- Generally unreliable, especially when representing a relatively small
- Population (can change radically with only a minor change in the distribution).
- Can be misleading; the mode tells you which score is most frequent, but
- Tells you nothing about the other scores in the distribution (radical changes can be
- Made to the distribution without changing the mode).
- Cannot be easily used in conjunction with inferential statistics.

**Limitations of Mean, Median and Mode :**

- The
**Mean**is the most stable measure of the central tendency, easy to understand and easy to calculate. It lakes all values of data into consideration. It is the best measure to estimate the population values from sample values. - The
**Median**is the middle most point in the distribution, also the best measure of central tendency and when extreme scores affect the mean, the best measure is the median and also when the measure of the distribution is open ended, i.e., when the lower limit of the lowest class interval and upper limit of the highest class interval is not known. But like the mean, the median cannot be subjected to mathematical operations. - The
**Mode**is the easiest measure of central tendency to calculate and understand. It can be identified by simple observation. It corresponds to the highest frequency of the frequency which occurred more frequently in the distribution mode also cannot be applied for mathematical operations like the median. Mode can be used with nominal, ordinal and interval scales of measures. Where mean and median can be used with interval or ratio scales of measurements.