**What is the Mode in Statistics**

Mode is also known as norm.

Mode is the value which occurs most frequently in a set of observations and around which the other items of the set cluster density.

**Algorithm**

**Step I : **Obtain the set of observations.

**Step II : **Prepare the frequency distribution.

**Step III : **Obtain the value which has the maximum frequency.

**Step IV : **The value obtained in step III is the mode.

The mode or model value of a distribution is that value of the variable for which the frequency is maximum. For continuous series, mode is calculated as,

**Symmetric distribution:** A distribution is a symmetric distribution if the values of mean, mode and median coincide. In a symmetric distribution frequencies are symmetrically distributed on both sides of the centre point of the frequency curve.

A distribution which is not symmetric is called a skewed-distribution. In a moderately asymmetric distribution, the interval between the mean and the median is approximately one-third of the interval between the mean and the mode i.e., we have the following empirical relation between them,

Mean – Mode = 3(Mean – Median)

⇒ Mode = 3 Median – 2 Mean.

It is known as **Empirical relation**.

**Relative characteristics of mean, median and mode**

- Mean is usually understood as arithmetic average, since its basic definition is given in arithmetical terms.
- Mean is regarded as the true representative of the whole population since in its calculation all the values are taken into consideration. It does not necessarily assume a value that is the same as one of theoriginal ones (which other averages often do).
- Mean is suitable for sets of data which do not have extreme values. In other cases, median is the appropriate measure of location.
- Mode is the most useful measure of location when the most common or most popular item is required.

** Read More:**

- How are Bar Graphs and Histograms Related
- Bar Graph in Statistics
- Median of Grouped Frequency Distribution
- Mean and its Advantages and Disadvantages
- Pie Charts
- Frequency Polygon

**Merits of Mode**

- Mode is readily comprehensively and easy to calculate. It can be located in some cases morely by inspection.
- Mode is not all affected by extreme values.
- Mode can be coneniently even class interval of unequal magnitude.

**Demerits of Mode **

- Mode is ill defined. In some cases we may come across two modes.
- It is not based upon all the observations.
- No further mathematical treatment is possible in case of mode.
- Mode is affected to a greater extant by flutuations of sampling.

**Relationship among Mean, Median and Mode :**

Following are the relations,

- Mode = 3 Median – 2 mean
- Median = Mode + (Mean – Mode
- Mean = Mode + (Median – Mode)

**Mode in Statistics Example Problems with Solutions**

**Example 1:** Find the mode from the following data :

110, 120, 130, 120, 110, 140, 130, 120, 140, 120.

**Solution: **Arranging the data in the form of a frequency table, we have

Value | Tally bars | Frequency |

110 | | | | 2 |

120 | | | | | | 4 |

130 | | | | 2 |

140 | | | | 2 |

Since the value 120 occurs maximum number of times i.e. 4. Hence, the modal value is 120.

**Example 2:** Find the mode for the following series :

2.5, 2.3, 2.2, 2.2, 2.4, 2.7, 2.7, 2.5, 2.3, 2.2, 2.6, 2.2

**Solution: **Arranging the data in the form of a frequency table, we have

Value | Tally bars | Frequency |

2.2 | | | | | | 4 |

2.3 | | | | 2 |

2.4 | | | 1 |

2.5 | | | | 2 |

2.6 | | | 1 |

2.7 | | | | 2 |

We see that the value 2.2 has the maximum frequency i.e. 4.

So, 2.2 is the mode for the given series.

** Example 3:** Compute mode for the following data.

7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 13, 13

**Solution: **Here, both the scores 8 and 10 occurs thrice (maximum number of times). So, we apply the empirical formula.

Here,

mean =

= =

=9.73

No. of scores = 15 (odd)

∴ Median = = t_{8} = 10

∴ Mode = 3 median – 2 mean

= 3 × 10 – 2 × 9.73 = 30 – 19.46 = 10.54

**Example 4:** Find the mode of the following data :

6, 4, 7, 4, 5, 8, 4, 5, 5, 3, 2, 5

**Solution: **We write the data in tabular form :

x | f |

2 | 1 |

3 | 1 |

4 | 3 |

5 | 4 |

6 | 1 |

7 | 1 |

8 | 1 |

We observe that 5 has maximum frequency which is 4

⇒ Mode = 5

**Example 5:** The following table gives the weights of 40 men. Calculate mode.

Weights (in kg) | Number of men |

54 | 6 |

72 | 6 |

80 | 1 |

64 | 2 |

62 | 6 |

60 | 5 |

58 | 5 |

56 | 4 |

63 | 5 |

**Solution: **Here, each of the scores 54, 72 and 62 occurs maximum number of times (six times). So we apply the empirical formula.

We construct the following table :

Weights x | No. of men f | Cumulative frequency | Product f.x |

54 | 6 | 6 | 324 |

56 | 4 | 10 | 224 |

58 | 5 | 15 | 290 |

60 | 5 | 20 | 300 |

62 | 6 | 26 | 372 |

63 | 5 | 31 | 315 |

64 | 2 | 33 | 128 |

72 | 6 | 39 | 432 |

80 | 1 | 40 | 80 |

Total | 40 | 2465 |

Mean = = = 61.625

Here, No. of scores = 40 (even)

Median = = = 61

∴ Mode = 3 median – 2 mean

= 3 × 61 – 2 × 61.625

= 183 – 123.25 = 59.75

Thus, modal weight = 59.75 kg

**Example 6:** If mean = 60 and median = 50, find mode.

**Solution: **We have,

Mean = 60, Median = 50

Mode = 3 Median – 2 Mean

= 3 (50) – 2 (60) = 30

**Example 7:** If mode = 70 and mean = 100, find median.

**Solution: **We have, Mode = 70, Mean = 100

Median = Mode + (Mean – Mode)

= 70 + (100 – 70)

= 70 + 20

= 90

**Example 8:** If mode = 400 and median = 500, find mean.

**Solution: **Mean = Mode + (Median – Mode)

= 400 + (500 – 400)

= 400 + (100)

= 400 + 150

= 550

**Example 9:** Find the mode of the data 3, 2, 5, 2, 3, 5, 6, 6, 5, 3, 5, 2, 5.

**Solution: **Since 5 is repeated maximum number of times, therefore mode of the given data is 5.

**Example 10:** If the value of mode and mean is 60 and 66 respectively, then find the value of median.

**Solution: **Mode = 3 Median – 2 mean

∴ Median = (mode + 2 mean)

= (60 + 2 × 66) = 64