**Class 11 statistics Chapter 5 Measures of central tendency**

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**NCERT Notes for Class 11 statistics Chapter 5 Measures of central tendency**

**Class 11 statistics Chapter 5 Measures of central tendency**

There are several statistical measures of central tendency or “averages”.

The three most commonly used averages are:

**Arithmetic Mean****Median****Mode**

**ARITHMETIC MEAN**

- Arithmetic mean is the most commonly used measure of central tendency.
- It is defined as the sum of the values of all observations divided by the number of observations and is usually denoted by X.

*Two interesting properties of A.M**. *

- The sum of deviations of items about arithmetic mean is always

equal to zero. Symbolically, Σ (X – X) = 0.

- Arithmetic mean is affected by extreme values. Any large value, on either end, can push it up or down.

**Weighted Arithmetic Mean:** Sometimes it is important to assign weights to various items according to their importance when calculating arithmetic mean. It is called Weighted Arithmetic Mean.

**MEDIAN**

- Median is that
*positional value*of the variable which divides the distribution into two equal parts, one part comprises all values greater than or equal to the median value and the other comprises all values less than or equal to it. - The Median is the “middle” element when the data set is arranged in order of the magnitude.

**Quartiles**

Quartiles are the measures which divide the data into **four equal parts**; each portion contains equal number of observations. There are three quartiles.

- The first Quartile (denoted by Q1) or lower quartile has 25% of the items of the distribution below it and 75% of the items are greater than it.
- The second Quartile (denoted by Q2) or median has 50% of items below it and 50% of the observations above it.
- The third Quartile (denoted by Q3) or upper Quartile has 75% of the items of the distribution below it and 25% of the items above it.
- Thus, Q1 and Q3 denote the two limits within which central 50% of the data lies.

**Percentiles**

Percentiles divide the distribution into hundred equal parts. It is denoted by P1, P2, P3, …, P99.( **P50 is the median value)**.

**MODE**

- The word mode has been derived from the French word
*“la Mode”*which signifies the most fashionable values of a distribution, because it is repeated the highest number of times in the series. - Mode is the most frequently observed data value. It is denoted by Mo.

**RELATIVE POSITION OF ARITHMETIC MEAN, MEDIAN AND MODE**

If Arithmetic Mean = Me

Median = Mi

Mode = Mo

- The relative magnitude of the three are
**Me>Mi>Mo or Me<Mi<Mo**(suffixes occurring in alphabetical order). - The
**median**is always between the arithmetic mean and the mode - β Measures of central tendency or averages are used to summarise the data. It specifies a single most representative value to describe the data set.
- β Arithmetic mean is the most commonly used average. It is simple to calculate and is based on all the observations. But it is unduly affected by the presence of extreme items.
- β Mode is generally used to describe the qualitative data. Median and mode can be easily computed graphically. In case of open-ended distribution, they can also be easily computed.