Introduction to Median of Ungrouped Data for Kids
The Median of Ungrouped Data is when you have numbers in a list and have to find the middle number. Usually, if there are an even number of numbers, the median is the average of the two in the middle.
Example of Ungrouped Data
The median of ungrouped data is calculated by finding which number falls at the exact centre point in an ordered set. This means that half (or exactly 50% or 0.5) of your data will be on one side and half on the other. Let's see in detail about median and median of ungrouped data with frequency
What is the Median Formula for Ungrouped Data?
Firstly, we have to arrange the given data in order to use the median formula.
Step 1. Arrange the given values in ascending order.
Step 2. Find the number of observations in the given set of data. It is denoted by $n$.
Step 3. If $n$ is odd, the median equals the $\dfrac{(n+1)}{2}^{\text {th }}$ observation.
Step 4. If $n$ is even, then the median is given by the mean of $\dfrac{n}{2}^{\text {th }}$ observation and $[\dfrac{n}{2}+1]^{\text {th }}$ observation.
Median Problems and Median Solutions
To calculate the Median Of Ungrouped Data, data needs to be put in order. Also, one must first group them and find the total number of items in a set. The next step is to add 1 and divide this total by two; the answer is the data point.
i.e., Median = Value of the (\[\frac{n+1}{2}\])\[^{th}\]
Solved Examples
Q 1 The heights (in cm) of 11 players of a team are as follows: 173, 158, 158, 159, 160, 160, 165, 166, 171, 170, 163.
Ans: On arranging the variates in ascending order, we get 158, 158, 159, 160, 160, 163, 165, 166, 170, 171, 173
There are 11 variates, and the number is odd.
Therefore, median = $\left[\dfrac{(11+1)}{2}\right]^{th}$ variate = 6th variate = 163.
Q 2 The median of 17, 13, 10, 15, x is an integer x then find x.
Ans: There are five variates. So, $\left[\dfrac{(5+1)}{2}\right]^{th}$ variate, is 3rd variate when written in ascending order will be the median x.
On arranging the variates in ascending order, we get 10, 13, x, 15, 17.
Therefore, 13 < x < 15.
But x is an integer. So, x = 14.
Q 3 Find the median of the first ten even numbers
Ans: First ten odd integers = 2,4,6,8,10,12,14,16,18,20
The number of variates = 10
Since n is even, we take two numbers in the middle, add them, and then divide by 2.
So, 10+12=22. We get 11 by dividing 22 by 2. So, the median is 11.
Q 4 Find the median of 1,2,3,4,5.
Ans: n= 5
Since the number of variates is odd, we have to add 1 and divide by 2. Now, n=5, on adding 1, it is 6. On dividing 6 by 2, we get 3, which is the data point. 3 falls on the 3rd point. So, the median is 3.
Q 5 Find the median of the following: 16, 24, 8, 12, 19
Ans: On arranging the data in ascending order, we get 8, 12, 16, 19, 24
There are 5 variates; the number is odd.
Therefore, median = $\left[\dfrac{(5+1)}{2}\right]^{th}$ variate = 3rd
Median = 16
Practice Questions
Q1. Find out the median of the following ungrouped data
1,3,9,5,3,7,9,2,6,9,5,3
Ans: 5
Q2. Find the Median: 36, 44, 86, 31, 37, 44, 86, 35, 60, 51
Ans: 44
Q3. The median of observations 11, 12, 14, 18, x + 2, x + 4, 30, 32, 35, 41 arranged in ascending order is 24. Find the values of x.
Ans: x = 21
Q4. Find whether the data is grouped or Ungrouped. The ages of children at a music show are as follows: 10, 9, 10, 11, 12, 8, 8, 9, 9
Ans: Ungrouped data
Q5. State whether the following is True or False. A list of the heights of every student in a class would be an example of data.
Ans: True
Summary
The median of ungrouped data helps to find the middle number in an ordered list or the average of two numbers that are put together. For example, if you have 5 numbers and your data looks like this 5, 6, 7, 8, 10, then you would calculate your mean as 5 because it uses all of your data. Then take these values and divide them by two and one to get your median of 4.6 because it used half of this group and one-half from another group.
FAQs on Median of Ungrouped Data
1. What is the median of ungrouped data and how is it calculated?
The median is the value of the middle-most observation in a dataset that has been arranged in ascending or descending order. For ungrouped data, the calculation method depends on the number of observations (n):
- If n is odd: The median is the value of the ((n+1)/2)th observation.
- If n is even: The median is the average of the values of the (n/2)th and ((n/2) + 1)th observations.
2. Can you provide an example of calculating the median for both an odd and an even number of data points?
Certainly. Let's look at both cases:
- Odd number of data points: Consider the data set {3, 9, 4, 1, 7}. First, arrange it in ascending order: {1, 3, 4, 7, 9}. There are 5 observations (n=5). The median is the ((5+1)/2)th or 3rd value, which is 4.
- Even number of data points: Consider the data set {10, 2, 8, 4}. First, arrange it: {2, 4, 8, 10}. There are 4 observations (n=4). The median is the average of the (4/2)th (2nd) and ((4/2)+1)th (3rd) values. So, Median = (4 + 8) / 2 = 6.
3. What is the main difference between finding the median for ungrouped data versus grouped data?
The primary difference lies in the precision and method. For ungrouped data, you identify the exact middle value(s) from an ordered list of individual observations. For grouped data (data in class intervals), you cannot find the exact median. Instead, you first identify the 'median class' where the middle observation lies and then use a specific formula to estimate the median's value within that interval.
4. Why is the median sometimes a better measure of central tendency than the mean?
The median is often preferred over the mean when the dataset contains outliers or extreme values. The mean is sensitive to these outliers and can be skewed, giving a misleading picture of the 'center' of the data. The median, being a positional average, is not affected by extreme values. For example, in calculating the average salary of a team, one very high salary would significantly inflate the mean, whereas the median would provide a more realistic representation of the typical salary.
5. How do you find the median when ungrouped data is presented in a frequency distribution table?
When data is in a frequency distribution table, you must first calculate the cumulative frequency (cf). Then, find the total number of observations (N). The position of the median is determined using the (N+1)/2 or N/2 rules. You then locate the observation in the table whose cumulative frequency is equal to or just greater than this position. That observation is the median of the dataset.
6. Why must data be arranged in ascending or descending order before finding the median?
Arranging data in order is a critical first step because the very definition of a median is based on its position. The median is the value that splits the dataset into two equal halves. This 'middle' position is only meaningful and correct when the data is sorted. If the data is not ordered, the value in the physical middle of the list would be arbitrary and would not accurately represent the central tendency of the data.
7. What are the limitations of using the median for statistical analysis?
While useful, the median has some limitations:
- It does not consider the magnitude of all values in the dataset, only the middle one(s).
- It is not capable of further algebraic treatment. For instance, you cannot find a combined median of two datasets simply from their individual medians, unlike the mean.
- It can be more time-consuming to calculate for large, unordered datasets as it requires sorting the entire list first.
