Introduction: Statistics

Ungrouped data- 

Ungrouped data is data in its original or raw form. The observations are not classified into groups. For example, the ages of everyone present in a classroom of kindergarten kids with the teacher is as follows:

3, 3, 4, 3, 5, 4, 3, 3, 4, 3, 3, 3, 3, 4, 3, 27

Grouped data –

In grouped data, observations are organized in groups. For example, a class of students got different marks in a school exam. The data is tabulated as follows:

This shows how many students got the particular mark range. Grouped data is easier to work with when a large amount of data is present.

Frequency –

Frequency is the number of times a particular observation occurs in data.

Class interval-

Data can be grouped into class intervals such that all observations in that range belong to that class.

Class interval = upper class limit – lower class limit

Mean-

The mean is the mathematical average of a set of two or more numbers. The given below are the steps to find out mean-
Step 1: Classify the data into intervals and find the corresponding frequency of each class.

Step 2: Find the class mark by taking the midpoint of the upper and lower class limits.

Step 3: Tabulate the product of the class mark and its corresponding frequency for each class. Calculate their sum

Step 4: Divide the above sum by the sum of frequencies to get the mean.

Assumed mean method for finding out mean-

Step 1: Classify the data into intervals and find the corresponding frequency of each class.

Step 2: Find the class mark by taking the midpoint of the upper and lower class limits.

Step 3: Take one of the xi’s (usually one in the middle) as the assumed mean and denote it by ′a′.

Step 4: Find the deviation of ′a′ from each of the x′is  di=xi−a

Step 5: Find the mean of the deviations

Step 6:  Calculate the mean as:

Step deviation method for finding out mean-

Step 1: Classify the data into intervals and find the corresponding frequency of each class.

Step 2: Find the class mark by taking the midpoint of the upper and lower class limits.

Step 3: Take one of the x′is (usually one in the middle) as assumed mean and denote it by ′a′.

Step 4: Find the deviation of a from each of the x′is  di=xi−a

Step 5: Divide all deviations −di by the class width (h) to get u′is  ui=xi−a/h

Step 6: Find the mean of u′is

Step 7:  Calculate the mean as:

Median-

The median is the value separating the higher half of a data sample, a population or a probability distribution from the lower half. In simple terms, it may be thought of as the “middle” value of a data set.

Finding the median of a grouped data when class interval is not given-

Step 1: Tabulate the observations and the corresponding frequency in ascending or descending order.

Step 2: Add the cumulative frequency column to the table by finding the cumulative frequency up to each observation.

Step 3: If the number of observations is odd, the median is the observation whose cumulative frequency is just greater than or equal to (n+1)/2

If the number of observations is even, the median is the average of observations whose cumulative frequency is just greater than or equal to n/2 and (n/2)+1.

Cumulative frequency- it is obtained by adding all the frequencies up to a certain point.

Finding the median of a grouped data when class interval is given-

Step 1: find the cumulative frequency for all class intervals.

Step 2: the median class is the class whose cumulative frequency is greater than or nearest to n/2,by counting the observations and using tally marks to construct a frequency table. The where n is the number of observations.

Step 3: Median = l + [(N/2 – cf)/f] × h

Where,
l = lower limit of median class,
n = number of observations,
cf = cumulative frequency of class preceding the median class,
f = frequency of median class,
h = class size (assuming class size to be equal).

Mode-

In grouped data without class intervals, the observation having the largest frequency is the mode.
For ungrouped data, the mode can be found out observation having the largest frequency is the mode.

Finding mode for a grouped data when class interval is given-

For, grouped data, the class having the highest frequency is called the modal class. The mode can be calculated using the following formula. The formula is valid for equal class intervals and when the modal class is unique.

Mode = l + [(f₁ – f₀)/(2f₁ – f₀ – f₂)] × h

Where,
l = lower limit of modal class
h = class width
f₁ = frequency of the modal class
f₀ = frequency of the class preceding the modal class
f₂ = frequency of the class succeeding the modal class