वर्गीकृत आँकड़ों का माध्यक

वर्गीकृत आँकड़ों का माध्यक निरंतर और आवृत्ति वितरण के रूप में आँकड़ों का माध्यक है। माध्यक दिए गए आँकड़ों का सबसे मध्यमान मान है जो आँकड़ों के निचले आधे भाग को ऊपरी आधे भाग से अलग करता है। वर्गीकृत आँकड़ों के माध्यक की गणना करते समय निम्नलिखित बातें उपस्थित होती हैं:

• माध्यिका वर्ग
• संचयी बारंबारता
• वर्गीकृत आँकड़ों के सूत्र की माध्यिका

माध्यिका की परिभाषा

Median is the middlemost value in a given data set after it is arranged in ascending order. If the total number of items in the list is odd, then after arranging the values in ascending order the middlemost value is taken as the median.

Median = ${\frac {(n+1)}{2}}$ ^th term, where $n$ is the total number of observations.

If the number of items in the data set is even, then the average of the two middle values is taken as the median.

Median = $({\frac {n}{2}}$ ^th term + $({\frac {n}{2}}+1)$^th term$)$/ $2$ where $n$ is the total number of observations.

Example: Let's consider the data: $48,20,50,69,73,85$. What is the median ?

Solution:

Arranging in ascending order, we get $20,48,50,69,73,85$. Here, $n$ (no.of observations) = $6$

To find the median of even data we use the formula:

Median = $({\frac {n}{2}}$ ^th term + $({\frac {n}{2}}+1)$^th term$)$/ $2$

Median = $({\frac {6}{2}}$ ^th term + $({\frac {6}{2}}+1)$^th term$)$/ $2$

Median = $(3$^rd term + $4$^th term$)$/ $2$

Median = ${\frac {(50+69)}{2}}={\frac {119}{2}}=59.5$

Median of Grouped Data Formula

Median = $l+\left[{\frac {({\frac {n}{2}}-c)}{f}}\right]\times h$

• $l$ = lower limit of median class
• $n$ = total number of observations
• $c$ = cumulative frequency of the preceding (of median class) class
• $f$ = frequency of median class
• $h$ = size of each class

Steps to Find Median of Grouped Data

Finding the median of any given data is simple since the median is the middlemost value of the data. Since the data is grouped, it is divided into class intervals. Here are the steps to finding the median of grouped data.

• Step 1: Construct the frequency distribution table with class intervals and frequencies.
• Step 2: Calculate the cumulative frequency of the data by adding the preceding cumulative value of the frequency with the current value.
• Step 3: Find the value of $n$ by adding the values in frequency (which is nothing but the last value of the cumulative frequency column)
• Step 4: Find the median class. If $n$ is odd, the median is the ${\frac {(n+1)}{2}}$^th value. If n is even, then the median will be the average of the ${\frac {n}{2}}$^th and the $({\frac {n}{2}}+1)$^th observation.
• Step 5: Find the lower limit of the class interval and the cumulative frequency.
• Step 6: Apply the formula for median in statistics: Median = $l+\left[{\frac {({\frac {n}{2}}-c)}{f}}\right]\times h$

Let us look at an example to understand this better.

Calculate the median for the following data:

Marks	Number of students
$0-20$	$6$
$20-40$	$20$
$40-60$	$37$
$60-80$	$10$
$80-100$	$7$

Solution:

We need to calculate the cumulative frequencies to find the median.

Marks	Number of students	Cumulative frequency
$0-20$	$6$	$0+6=6$
$20-40$	$20$	$6+20=26$
$40-60$	$37$	$26+37=63$
$60-80$	$10$	$63+10=73$
$80-100$	$7$	$73+7=80$

$n=$ last value of cumulative frequency $=80$

Since $n$ is even, we will find the the average of the ${\frac {n}{2}}$^th and the $({\frac {n}{2}}+1)$^th observation i.e. the cumulative frequency greater than $40$ is $63$ and the class is $40-60$. Hence, the median class is $40-60$.

$l=40,f=37,c=26,h=20$

Using the median formula.

Median = $l+\left[{\frac {({\frac {n}{2}}-c)}{f}}\right]\times h$

$=40+\left[{\frac {({\frac {80}{2}}-26)}{37}}\right]\times 20$

$=40+\left[{\frac {(40-26)}{37}}\right]\times 20$

$=40+\left[{\frac {14}{37}}\right]\times 20$

$=40+7.57$

Median $=47.57$