The following are the distinctions between Mean Median and Mode:

- The mean is the average of the given observations’ values.
- The median of the given observations is the value in the middle.
- The mode is the most frequently occurring value in the given observation.

## Central tendency:

The word central tendency itself represents that It is a central value of a statistical series or a representative value. It is descriptive statistics of continuous variables.

In continuous variables there are

- Measures of central tendency: Mean Median and Mode
- Measures of dispersion
- Distributional forensics

An average is a single value within the range of the data that is used to represent all of the values in the series. Such an average is somewhere within the range of the data, therefore called as measures of central tendency. It is also called the average value or measure of location.

Measures of central tendency are of three which are as follows: Mean Median and Mode

### Arithmetic Mean:

Mean is the sum of observations of data divided by the number of observations.

Mean = Sum of observations ∕ Number of observations

Or

x =𝚺x∕n

Mean is divided into

- Mean of raw data
- Mean of ungrouped frequency distribution
- Mean of ungrouped frequency distribution by deviation method

#### Mean of a raw data:

Example:

Rainfall of a place in a week is 4 cm, 5cm, 12cm, 3cm, 6cm, 8cm, 0.5cm. Find the average rainfall per day.

Solution:

The average rainfall per day is the arithmetic mean of the above observations.

Given rainfall through a week are 4cm, 5cm, 12cm, 3cm, 6cm, 8cm, 0.5cm.

Number of observations (n) = 7

Mean = 𝚺x ∕ n |

Mean = 4+5+12+3+6+8+0.5 ∕ 7= 5.5cm

Consider an example: weights of students in a class are given in the following frequency distribution table.

Weights in kg (x) | 30 | 32 | 33 | 35 | 37 | 41 |

No. of students (f) | 5 | 9 | 15 | 6 | 3 | 2 |

Mean (x) = sum of all observations ∕ total number of observations.

From the above table we can see that 5 students weigh 30kg; each. So sum of their weight is 5*30=150 kg. Similarly, we can find out the sum of weights with each frequency and their total. The Sum of the frequencies gives the number of observations in the data.

So mean =

5×30+ 9× 32+ 15×33 + 6×35 + 3×37 + 2×41 ∕ 5+9+15+6+3+2 =33.4 kg

x= 𝚺 fx ∕ 𝚺 f |

#### Mean of ungrouped frequency distribution:

This is again of 2 types:

- Simple method
- Deviation method

#### Simple method:

Thus in the case of ungrouped frequency distribution, you can use the formula,

x= 𝚺 fx ∕ 𝚺 f |

#### Deviation method:

In this method, we assume one of the observations which are convenient as the assumed mean.

We can use formula,

Mean (x)= A + 𝚺 fd ∕ 𝚺 f |

### Median :

Median is the middle observation of a given raw data, when it is arranged in an order (ascending/ descending). it divides the data into two groups of equal numbers, one part comprising all values greater than the median and the other part comprising values less than the median.

**When the data has an ‘n’ number of observations and if ‘n’ is odd, the median is (n+½)th observation. **

**When n is even, the median is the average of (n/2)th and (n/2+1)th observations.**

### Mode:

Mode is the value of the observation which occurs most frequently,** i.e; an observation with the maximum frequency is called mode.**

Example:

The following numbers are the sizes of shoes sold by a shop in a particular day. Find the mode?

6, 7,8,9,10,6,7,10,7,6,7,9,,6

Solution:

First, we have to arrange the observation in order

6,6,6,6,7,7,7,7,7,8,9,9,10,10 to make frequency distribution table

Size | 6 | 7 | 8 | 9 | 10 |

No. of shoes sold | 4 | 5 | 1 | 2 | 2 |

Here no.7 occurred most frequently i.e; 5 times.

∴ The mode of the given data is 7. This indicates the shoes of size no.7 are fast-selling items.

Let’s summarize the difference between Mean Median and Mode

## Difference between Mean Median and Mode:

mean | median | mode |
---|---|---|

Meaning | ||

Mean is the average of the value in a series of data. | Median is the middlemost value in a series of data. | Mode is the most occurring value in a series of data. |

Type of Average | ||

It is a mathematical average | It is a positional average. | Mode is a positional average. |

Basis | ||

Mean is based on all observations | Median is the middlemost value | Mode is a most occurring item |

Capability | ||

Mean is capable of further algebraic treatment. | Median is not capable. | Mode is also not capable. |

Observation | ||

Mean can be obtained only by calculation. | Median can be obtained by mere observation. | It also can be obtained by mere observation. |

Location | ||

Mean cannot be located in graph | Median can be located on the graph | Mode also can be located graphically. |

Affected by | ||

These are affected by extreme values. | Median is not much affected by extreme values. | Mode is also not much affected by extreme values. |

Defined | ||

Mean is well defined in all cases. | Median is also well defined in all cases. | Mode is ill-defined in some cases. |

Usage | ||

It cannot be used when: percentage. 1. Distribution is highly skewed. 2. Distribution has open end classes. 3. Avg. required is for rates, ratios and | Median is suitable when: 1. The data are not capable of direct measurement. 2.The distribution is of open-end classes. | Mode is very often ill-defined 1.Mode is used in problems involving the expression of preferences. |

## Bottom line:

As a result of the preceding discussion, The comparision of Mean Median and Mode the majority of the time, we want to calculate the central tendency of a dataset. The mean is preferred over the other two entities when working on a dataset to measure central tendency because it takes into account every single value in the dataset.

Depending on the situation, you can select the mean median and mode that best represents the data sections that you’re identifying.

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