Statistics: Mean, Median, Variance and Cumulative Frequency Curve

Statistics is a branch of mathematics that is focused on collecting, analyzing, organizing, and interpreting different types of data. It holds immense value in scientific, economic, social, and industrial studies.

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Types of Data

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• Ungrouped Data:

The data which is present in its original or unsorted form is called ungrouped data. It is the set of data that you initially get from a survey or experiment.

Example: Let’s say the marks of 15 students present in a class are:

18, 14, 8, 10, 18, 19, 16, 15, 18, 7, 15, 19, 10, 17, 12

From this data, we can interpret that 19 is the highest mark and two students have got the highest marks. It is easy to work on ungrouped data if the number of data is small.

• Grouped Data:

The data which is sorted into different categories or groups is called grouped data.

For example, we have the marks obtained by 50 students in an examination. Such huge data is difficult to interpret in an ungrouped manner. So we sort the data as given in the table below:

• Frequency

The total number of times that a specific observation is present in data is called frequency. 

• Class interval

Class interval is used to group the observations into specific classes. Considering the table given above, we have grouped the number of students into the marks interval. Here, the marks interval is the class interval. 

Class width= upper limit of class - the lower limit of class

Mean, Median and Mode Video Explanation

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Measures Of Central Tendency

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Mean

In simple terms, the average of a data set is called the mean.

For ungrouped data, 

Mean = sum of observations÷ total number of observations

Mean= 

x=x1 + + x2 + x3+ x4+……+ xnn

For grouped data, there are three methods of finding the mean. 

• Direct method:

X=∑fixi/fi

Where fi= frequency of ith class with a class mark of xi.

Class mark: Class mark is the midpoint of a class that contains the frequency.

Class mark =(upper limit of class + lower limit of class)/2

• Assume mean method:

In this method, we have to select one of the numbers generally present in the middle, as a supposed mean, 'a'.

We have to find the value of 'd' that is the deviation of 'a' from all the observations xi. 

• Step deviation method:

Find the class mark of the class interval by taking a point of lower and upper-class limits. (Formula of classmark is mentioned ready)

Now take any value of xi, preferably from the middle, and represent it with “a”.

With the formula given below, find the deviation of a:

di=xi-a

Now you have to find out the mean of the deviations with the formula given below:

d? =∑xidi/∑fi

Finally, calculate the mean:

x? = a + ∑xidi/∑fi

Median

The middlemost observation of a set of data is called the median. Methods of finding median are:

• Median of grouped data without class intervals:

• Arrange all the observations and frequencies in ascending order. 
• Add all the frequencies up to a certain observation to find the cumulative frequency of that observation. 
• For an odd number of observations, the median is the observation whose cumulative frequency is just larger than or equal to (n+1)/2.
• The median is the average of observations whose cumulative frequency is just larger than or equal to n/2 and (n/2)+1, in the case of an even number of observations.

• Median of grouped data with class interval:

It is not possible to find the median of grouped data by observing the frequency table, so we have to use the formula given below:

Median= l + (n/2-cf )/f X h

Where,

n= number of observations,

l= the lower limit of median class,

cf=the cumulative frequency of the class that precedes the median class,

h= class width,

f= frequency of median class

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Graphical method of finding median:

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To find the median through the graph, first, you have to draw the graph properly. Follow the steps below:

1.The first step of finding the median is to determine the median class.

2.Label the y-axis with cumulative frequencies and the x-axis with data belonging to the median class.

3.Connect the extremes of class and cumulative frequencies with a straight line graph.

4.Locate the graph point that corresponds to cf=n/2.

5.From this point on t, drop a perpendicular.

A cumulative frequency curve, also known as an ogive, is created by linking the points.

Ogives are divided into two categories:

• Less Than Ogive: Cumulative frequencies of 'less than' are displayed against upper-class limits of distinct class intervals. It is usually an ascending curve.

•  More Than Ogive: The cumulative frequencies of 'more than' are displayed against the lower class limits. It is usually a decreasing curve.

Find the point at which both ogive types intersect and this point denotes the median of the grouped data.

Mode

The observation or number which occurs most often or frequently in a data set is called mode. 

•  Mode of ungrouped data:

Finding the mode of ungrouped data is very easy. Just notice which observation has the maximum frequency and that observation is considered to be the mode of that particular set of data.

For example, consider the numbers:

7, 8, 18, 7, 3, 20, 3, 18, 7

The number 7 occurs most frequently, therefore, mode=7.

• Mode of Grouped Data:

You can find the mode of grouped data using the following formula:

l=(f₁-f₀/2f₁-f₀-f₂)X _h

where l= lower limit of modal class,

h= class width,

f₀= frequency of the class that precedes the median class

f₁= frequency of the modal class,

f₂= frequency of the class that succeeds the modal class.