Mode Lecture 08

 Mode 

Lecture 08

Mode

Mode is a French word that means fashion. The thing that is very common in our daily lives is called fashion. Mode is also a central measure of data. Mode is an important measure of central tendency and is used in some specific situations.

For example, 

* The educational institutions to determine the most common learning challenges that students face.

* The health care professionals use mode to identify the most common disease in a particular area or season.

* The government uses mode to identify the most common crops and livestock. 

Definition

The most frequent value or values in a dataset.

In gruoped data mode, it can be obtained by the following formula:

Where:

h = L2 - L1

L1 & L2 are the lower and upper class boundaries, respectively.

Example 3. 26: The score of a student in the last 10 tests is given below:

8, 6, 10, 9, 8, 5, 7, 8, 4, 6.

What is the model score?

Solution: Just by looking at the data, we find the frequency of 8 is 3 and more than the frequency of all others.

So, mode of the data is 8.

Example 3.27: Estimate mode for the following frequency distribution.

Classes	1 – 5	6 – 10	11 - 15	16 – 20	21- 25
Frequency	2	7	8	3	1

Solution:

Classes	Class boundaries	Frequency
1 – 5	0.50 – 5.50	2
6 – 10	5.50 – 10.50	7
11 – 15	10.50 – 15.50	8
16 – 20	15.50 – 20.50	3
21- 25	20.50 – 25.50	1

The class 10.50–15.50 has maximum frequency, so it is called a model class. The frequency of the model class is denoted by fm.

Mode can be obtained from Histogram

i. Identify the tallest bar. This represents the modal class.

ii. Join the tips of this to those of the neighboring bars on either side, with the one on the left joined to that on the right. The lines used to join these tips cross each other at some point in this bar.

iii. Drop a perpendicular line from the tip of the point where these lines meet to the base of the bar. The point where it meets the base is the mode.

Example 3.28: Construct a histogram graphically for the following frequency distribution.

Mark	10 – 19	20 – 29	30 – 39	40 – 49	50 – 59	60-69	70 – 79
No. of students	10	12	18	30	16	6	8

Solution:

Marks	Class boundaries	Frequency
10 – 19	9.50 – 19.50	10
20 – 29	19.50 – 29.50	12
30 – 39	29.50 – 39.50	18
40 – 49	39.50 – 49.50	30
50 – 59	49.50 – 59.50	16
60-69	59.50 – 69.50	6
70 – 79	69.50 – 79.50	8

Mode = 44.12 

Advantages and limitations of Mode.

Some advantages of mode are given below:

i. It is easy to calculate.

ii. It is not affected by extreme values.

iii. It can be useful for qualitative data.

iv. It can be located graphically.

Some limitations of mode are given below:

i. It is not based on all the values.

ii. It is not well defined.

iii. Its capability for further mathematical treatment is limited.

• Read: Median

Harmonic Mean Lecture 07

 Harmonic Mean 

Lecture 07

Introduction

The harmonic mean is a measure of central tendency and very useful in averaging rates and ratios types of data, such as price (Rs./kg), speed (km/hour), and productivity (output/manhour). The harmonic mean is applicable in the following situations:.

i. When ratio is expressed as x per y, if x is given, use hormonic, and if y is given, then use arithmatic mean. 

ii. The data have a few points that are much higher than the rest.

Definition

The reciprocal of the arithmatic mean of the reciprocal of values.

Let X1, X2,..., Xn be the values of a data. The harmonic mean is obtained as:

When the data is expressed in the frequency distribution, then the harmonic mean can be obtained as:

Example 3.18: The distance from residential area to market is 40 km. A driver drove to a residential area at a speed of 50 km per hour and returned to the market at a speed of 80 km per hour. What is the average speed for the whole journey?

Solution: As the data is expressed as per (50/per hour) and x is given, then we use harmonic mean.

Example 3.19: A bus traveled 40 km in four segments at a speed of 100 km per hour for the first 10 km, 110 km per hour for the second 10 km, 90 km per hour for the third 10 km, and 120 km per hour for the fourth 10 km. What is the average speed of the bus?

Solution:

Number	Distance	Speed (Km per hour)	Time (hour)
1	10	100	0.100
2	10	110	0.091
3	10	90	0.111
4	10	120	0.083
Total	40		0.385

The harmonic mean formula is:

Example 3.20: Find the harmonic mean for the data given below:

10, 16, 20, 25, 28, 32, 46.

Solution: The harmonic mean for a discrete set of data is given below:

Example 3.21: Find the harmonic mean for the following frequency distribution.

Absentees	2	3	4	5	6	7	8	9
Number of students	15	10	16	20	21	15	9	7

Solution:

Example 3.22: Find the hamonic mean for the following age distribution.

Age	10 - 20	20 - 30	30 – 40	40 – 50	50- 60	60 - 70
No. of person	50	80	145	95	60	55

Solution:

Merits and Demerits of Harmonic Mean

Merits of harmonic mean

i. It is rigorously defined by proper mathematical formula.

ii. It is based on all the observations of the data.

iii. It is suitable average in case series having some observation very large.

iv. It is a suitable average for rates and ratios.

Demerits of harmonic mean

i. It is not easy to understand and interpret.

ii. It gives too much weight to smaller observations of the data.

iii. It cannot be computed if any value in the data is zero.

iv. It cannot represent distribution.

Relationship between Mean, Geometric Mean and Harmonic Mean

i. If all the observations are the same, the arithmetic mean, geometric mean, and harmonic mean are equal.

ii. If all the observations are not the same, then mean will be greater than geometric mean and geometric mean will be greater than harmonic mean.

iii. If there are two values, then the square of the geometric mean is equal to the product of arithmetic and harmonic means.

Example 3.23: The arithmetic and geometric means of two values are 25 and 23.50, respectively. Find the harmonic mean.

Solution: The mean and G.M. of two values are given. Now to find the H.M., we use the following relation:

Example 3.24: Compare the arithmetic mean, geometric mean, and harmonic mean of the following data.

10, 16, 20, 25, 32.

Solution:

• Read: Mode