unit 1 short notes

Syllabus

Measure of Central Tendency: Mean, Median and Mode; Dispersion: Range, Quartile, Percentile, Quartile Deviation, Mean Deviation, Standard Deviation, Coefficient of Variation and Variance; Skewness: Definition, Importance and Measures of Skewness.

Measure of Central Tendency: Mean, Median and Mode; 

INTRODUCTION:

The objective here is to find one representative value which can-be used to locate and summarise the entire set of varying values. This one value can be used to make many decisions concerning the entire set. We can define measures of central tendency (or location) to find some central value around which the data tend to cluster. 

SIGNIFICANCE OF MEASURES OF CENTRAL TENDENCY:

Measures of central tendency i.e. condensing the mass of data in one single value , enable us to get an idea of the entire data. For example, it is impossible to remember the individual incomes of millions of earning people of India. But if the average income is obtained, we get one single value that represents the entire population. Measures of central tendency also enable us to compare two or more sets of data to facilitate comparison. For example, the average sales figures of April may be compared with the sales figures of previous months

PROPERTIES OF A GOOD MEASURE OF CENTRAL TENDENCY

[1] It should be easy to understand and calculate. 

[2] It should be rigidly defined. 

[3] It should be based on all observations. 

[4] It should be least affected by sampling fluctuation. 

[5] It should be capable of further algebraic treatment. 

[6] It should be least affected by extreme values. 

[7] It should be calculated in case of open end interval. 

Following are some of the important measures of central tendency which are commonly used in business and industry. 

 Arithmetic Mean 

 Weighted Arithmetic Mean 

 Median 

 Quantiles(quartiles, deciles and percentiles) 

 Mode 

 Geometric Mean 

 Harmonic Mean

ARITHMETIC MEAN 

The arithmetic mean (or mean or average) is the most commonly used and readily understood measure of central tendency. 

In statistics, the term average refers to any of the measures of central tendency. 

Ungrouped data/Raw data 

The arithmetic mean is defined as being equal to the sum of the numerical values of each and every observation divided by the total number of observations. Symbolically, it can be represented as: 

where, 

∑X indicates the sum of the values of all the observations, and N is the total number of observations. 

For example, 

let us consider the monthly salary (Rs.) of 10 employees of a firm x 

2500, 2700, 2400, 2300, 2550, 2650, 2750, 2450, 2600, 2400 

If we compute the arithmetic mean, then 2500+2700+2400+2300+2550+2650+2750+2450+2600+2400 = 25300

Mean=25300 /10= Rs. 2530. 

Therefore, the average monthly salary is Rs. 2530. 

Discrete data

 When the observations are classified into a frequency distribution, Therefore, for discrete data; the arithmetic mean is defined as:

Where, f is the frequency for corresponding variable x and N is the total frequency, i.e. N = ∑f.

 

Mean=8270/200= 41.35 

Continuous Data 

When the observations are classified into a frequency distribution, Therefore, for grouped data; the arithmetic mean is defined as

Where X is midpoint of various classes, f is the frequency for corresponding class and N is the total             frequency, i.e. N =  ∑f. 

This method is illustrated for the following data which relate to the monthly sales of 200 firms. 

        

        the midpoint of the class interval would be treated as the representative average value of that                class. 

Mean=102000/200=510

MERITS OF MEAN 

[1] It is easy to understand and calculate. 

[2] It is rigidly defined. 

[3] It is based on all observations. 

[4] It is least affected by sampling fluctuation. 

[5] It is capable of further algebraic treatment. 

DEMERITS OF MEAN 

1) It is highly affected by extreme values. 

2) It is not calculated in case of open end interval.

MEDIAN 

A second measure of central tendency is the median. Median is that value which divides the distribution into two equal parts. 

Fifty per cent of the observations in the distribution are above the value of median and other fifty per cent of the observations are below this value of median. 

The median is the value of the middle observation when the series is arranged in order of size or magnitude (Ascending order). 

UNGROUPED DATA 

If the number of observations is odd, then the median is equal to one of the original observations (Middle).

example, if the income of seven persons in rupees is 1100, 1200, 1350, 1500, 1550, 1600, 1800, then 

Median =(7+1/2) = 4th value 

Median =1500 

If the number of observations is even, then the median is the arithmetic mean of the two middle observations. 

For example, if the income of eight persons in rupees is 

1100, 1200, 1350, 1500, 1550, 1600, 1800,1850, 

then the median income of eight persons would be 

1500+1550/2= 1525

DISCRETE SERIES 

First we find cumulative frequency.then locate (N+1/2) the value in cumulative frequency.corresponding that value of x is median.

CONTINUOUS DATA

Step 1: Make Cumulative Frequency (CF)

Cumulative frequency means:

👉 Keep adding frequencies one by one
👉 It tells you how many workers are covered till each class

Example:

Step 2: Find the position of the Median

Formula to find the position:

$$\frac{N+1}{2}=\frac{1000+1}{2}=\frac{1001}{2}=500.5$$

This means:

👉 The median value lies at the 500.5th worker
👉 Now we check in CF where 500.5 falls

CF values:

120, 245, 425, 585, 735, …

500.5 lies between:

• 425 (below)

• 585 (above)

So the median class is 35–40

Step 3: Apply the Median Formula

Median formula:

Put Values in Formula

$$\text{Median}=35+\left(\frac{500-425}{160}\right)\times 5$$

Step-by-step:

1️⃣ $500 - 425 = 75$

2️⃣ $\frac{75}{160} = 0.46875$

3️⃣ $0.46875 \times 5 = 2.34$

So:

$$\text{Median} = 35 + 2.34 = 37.34$$

Final Answer: Median = 37.34 years

MERITS OF MEDIAN 

[1] It is easy to understand and calculate. 

[2] It is rigidly defined. 

[3] It is not affected by extreme values. 

[4] It is calculated in case of open end interval. 

[5] It is located by graphically also. 

DEMERITS OF MEDIAN 

[1] It is not based on all observations. 

[2] It is affected by sampling fluctuation. 

[3] It is not capable of further algebraic treatment.

MODE 

The mode is the typical or commonly observed value in a set of data. It is defined as the value which occurs most often or with the greatest frequency. 

The dictionary meaning of the term mode is most usual'. 

For example, in the series of numbers 3, 4, 5, 5, 6, 7, 8, 8, 8, 9, the mode is 8 because it occurs the maximum number of times. 

That means in ungrouped data mode can find by inspection only. 

DISCRETE DATA 

    Mode is the value of X which has highest frequency. For example,

Mode=40 

CONTINUOUS DATA 

First, we find Modal class = corresponding to highest frequency.

Dispersion: Range, Quartile, Percentile, Quartile Deviation, Mean Deviation, Standard Deviation, Coefficient of Variation and Variance; Skewness: Definition, Importance and Measures of Skewness.