What is the Mode in Statistics

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

1. Mean is usually understood as arithmetic average, since its basic definition is given in arithmetical terms.
2. 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).
3. Mean is suitable for sets of data which do not have extreme values. In other cases, median is the appropriate measure of location.
4. 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

https://www.youtube.com/watch?v=hp6MEaYX9SU&t=24s

Merits of Mode

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

Demerits of Mode

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

Relationship among Mean, Median and Mode :
Following are the relations,

1. Mode = 3 Median – 2 mean
2. Median = Mode + \(\frac{2}{3}\) (Mean – Mode
3. Mean = Mode + \(\frac{3}{2}\) (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

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

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 = \(\frac{{7 \times 2 + 8 \times 3 + 9 \times 2 + 10 \times 3 + 11 \times 2 + 12 + 13 \times 2}}{{2 + 3 + 2 + 3 + 2 + 1 + 2}}\)
= \(\frac{{14 + 24 + 18 + 30 + 22 + 12 + 26}}{{15}}\) = \(\frac{{146}}{{15}}\)
=9.73
No. of scores = 15 (odd)
∴    Median = \({t_{\frac{{15 + 1}}{2}}}\) = t₈ = 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 :

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.

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 :

Mean = \(\frac{{\Sigma f.x}}{{\Sigma f}}\) = \(\frac{{2465}}{{40}}\) = 61.625
Here, No. of scores = 40 (even)
Median = \(\frac{{{t_{20}} + {t_{21}}}}{2}\) = \(\frac{{60 + 62}}{2}\) = 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 = \(\frac{1}{3}\) (mode + 2 mean)
= \(\frac{1}{3}\) (60 + 2 × 66) = 64