Table of contents  
Mean  
Median  
Mode  
Cumulative Frequency Chart  
Empirical Formula between Mode, Mean and Median 
The mean value of a variable is defined as the sum of all the values of the variable divided by the number of values.
The median of a set of data values is the middle value of the data set when it has been arranged in ascending order. That is, from the smallest value to the highest value
Median is calculated as:
Where n is the number of values in the data.
If the number of values in the data set is even, then the median is the average of the two middlevalue.
Mode of statistical data is the value of that variable that has the maximum frequency.
Here is the ungroup frequency table:
Mean is given by:
Greek letter ∑ (capital sigma) means summation.
In this distribution, it is assumed that the frequency of each class interval is centered around its midpoint i.e class marks.
(a) Direct Method
This method can be very calculationintensive if the values of f and x are large. We have big calculations and chance of making mistake is quite high
Steps involved in finding the mean using Direct Method
Example: The following table shows the weights of 10 children:
Find the mean by using the direct method.
Sol:
So, Mean would be
=698/10 = 69.8 kg.
(b) Assumed Mean Method
Where
a= Assumed Mean
d_{i} = x_{i} –a
This method is quite useful when the values of f and x are large. It makes the calculation easier. In this method, we take some assumed mean calculate the deviation from it and then calculate the mean using the above formula.
Steps involved in finding the mean using the Assumed Mean Method
Example: The following table shows the weights of 10 children:
Find the mean by using Assumed Mean method.
Sol:
Let the assumed mean = A = 71
So, Mean would be
= 7112/10 = 69.8 kg
c) Step Deviation Method
Where
a= Assumed mean
u_{i} = (x_{i} –a)/h
This method is quite useful when the values of f and x are large. It makes the calculation further easier by dividing the deviation from the common factor.
Steps involved in finding the mean using Step Deviation Method
Example: The following table shows the weights of 10 children:
Find the mean by using Step Deviation method.
Sol:
Let the assumed mean = A = 71 and h=2
So, Mean would be
=71+ (6/10) 2 = 69.8 kg
Modal Class: The class interval having highest frequency is called the modal class and Mode is obtained using the modal class
Mode formula is given as
Where
l = lower limit of the modal class,
h = size of the class interval (assuming all class sizes to be equal),
f_{1} = frequency of the modal class,
f_{0} = frequency of the class preceding the modal class,
f_{2} = frequency of the class succeeding the modal class.
Example: The following table shows the ages of the patients admitted in a hospital during a year
Find the mode.
Sol:
Modal class = 35 – 45, l = 35, class width (h) = 10, f_{1} = 23, f_{0} = 21 and f_{2} = 14
Substituting the values in the Mode formula given above we get
Mode= 36.8 year
The cumulative frequency of a class is the frequency obtained by adding the frequencies of all the classes preceding the given class.
Cumulative Frequency chart will be like
The above table cumulative frequency distribution of the less than type. We can similary make it like below
The table above is called a cumulative frequency distribution of the more than type.
How to find Median of a grouped data frequency table
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)
Example: A survey regarding the heights (in cm) of 60 girls of a school was conducted and the following data was obtained:
Find the median height.
Sol:
To calculate the median height, we need to find the class intervals and their corresponding frequencies.
The given distribution being of the less than type, 140, 145, 150, . . ., 165 given the upper limits of the corresponding class intervals. So, the classes should be below 140, 140  145, 145  150, . . ., 160  165. Observe that from the given distribution, we find that there are 4 girls with height less than 140, i.e., the frequency of class interval below 140 is 4 . Now, there are 11 girls with heights less than 145 and 4 girls with height less than 140. Therefore, the number of girls with height in the interval 140145 will be 114=7. Similarly, other can be calculated
So, n =60 and n/2=30 And cumulative frequency which is greater than and nearest to 30 is 40 , So median class 150155
l (the lower limit) = 150,
cf (the cumulative frequency of the class preceding 150  155) = 29,
f (the frequency of the median class 150  151) = 11,
h (the class size) = 5.
Now by Median Formula
= 150 + [(3029)/11]5
=150.45 cm
Empirical Formula between Mode, Mean and Median is given as 3 Median=Mode +2 Mean
We can represent Cummulative frequency distribution on the graph also. To represent the data in the table graphically, we mark the upper limits of the class intervals on the horizontal axis (xaxis) and their corresponding cumulative frequencies on the vertical axis (yaxis), choosing a convenient scale.
When we draw the graph for the cumulative frequency distribution of the less than type. The curve we get is called a cumulative frequency curve, or an ogive (of the less than type).
When we draw the graph for the cumulative frequency distribution of the more than type. The curve we get is called a cumulative frequency curve, or an ogive (of the more than type).
When we plot both these curve on the same axis, The two ogives will intersect each other at a point. From this point, if we draw a perpendicular on the xaxis, the point at which it cuts the xaxis gives us the median
115 videos478 docs129 tests

1. What is the formula to calculate the mean? 
2. How do you calculate the mode for grouped data? 
3. What is the cumulative frequency? 
4. How do you calculate the median for grouped data? 
5. What is the empirical formula that relates mode, mean, and median? 
115 videos478 docs129 tests


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