How to Find the Median
- Introduction
- Definition of Median
- Formula for Median
- Median Formula for Ungrouped Data
- Median Formula for Grouped Data
- Applications of Median
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
In any group, the median is the value in the middle. One of the simplest statistical measures to compute is the median. The data must be sorted in ascending order, to calculate the median, which is represented by the middle data point. As the median value is placed in the middle of the data points, it is easy to infer that half of the data points have values less than the median and half of the data points have values greater than the median.
In addition, the quantity of data points affects how the median is calculated. The median is the middle value when there are odd numbers of data points, and the average of the two middle values when there are even numbers of data points. Let's study more about medians and how they are calculated.
Definition of Median
Three of the most commonly used measures of central tendency are mean, median and mode. Median is one of them. The value that results from placing the observations in ascending order and calculating the middle observation is known as the data median. Many times, it is difficult to consider the complete date for representation. The median is helpful in situations where it is challenging to examine all of the data for representation. Median helps to represent a large number of data points with a single data point. Calculating the median is a simple task when it comes to statistical summary measures. The median is also termed as “Place Average” as the median is positioned in the center of a sequence.
Formula for Median
The middle value of an organized group of numbers can be determined using the median formula. The group's components must be listed in ascending order to determine this central tendency measure. The nature of the data (grouped or ungrouped) and the number of data points (odd or even) affect how the median formula is calculated. The following formulas can be used to determine the provided data's median.
Median Formula for Ungrouped Data
When calculating the median calculation for ungrouped data, we use the following steps:
Step `1`: Put the information in ascending order.
Step `2`: Then, add up all of the observations `('n')`.
Step `3`: Determine whether `'n'` is an odd or even number of observations.
When `n` is Odd
When the number of data points (\(n\)) is odd, the formula to calculate the median is as follows.
Given a dataset \(X\) with \(n\) observations, where \(n\) is an odd number, arranged in ascending order, the median (\(M\)) is simply the value of the middle observation.
If \(X = \{x_1, x_2, ..., x_n\}\), then the formula for finding the median when \(n\) is odd is:
\( M = x_{\frac{n+1}{2}} \)
Here, \(x_{\frac{n+1}{2}}\) represents the \(\frac{n+1}{2}\)th observation in the ordered dataset, which is the middle observation when \(n\) is odd.
When `n` is Even
When the number of data points (\(n\)) is even, calculating the median involves taking the average of the two middle values.
Given a dataset \(X\) with \(n\) observations, where \(n\) is an even number, arranged in ascending order, the median (\(M\)) is the average of the two middle observations.
If \(X = \{x_1, x_2, ..., x_n\}\), then the formula for finding the median when \(n\) is even is:
\( M = \frac{x_{\frac{n}{2}} + x_{\frac{n}{2}+1}}{2} \)
Here, \(x_{\frac{n}{2}}\) and \(x_{\frac{n}{2}+1}\) represent the \(\frac{n}{2}\)th and \(\frac{n}{2}+1\)th observations in the ordered dataset, respectively. These are the two middle values when \(n\) is even, and their average gives us the median.
Median Formula for Grouped Data
When data is organised into intervals or groups, we call it as grouped data. For grouped data, the calculation of the median is slightly different. Here's the formula for finding the median for grouped data:
Step `1`: Identify the Median Class: First, determine the class interval (group) that contains the median value. This is typically the interval where the cumulative frequency crosses \( \frac{n}{2} \), where \( n \) is the total number of observations.
Step `2`: Calculate the Median: Once the median class is identified, you can use the following formula to calculate the median (\( M \)):
\( M = L + \frac{\frac{n}{2} - F}{f} \times w \)
Where:
- \( L \) `=` lower limit of the median class
- \( n \) `=` total number of observations
- \( F \) `=` cumulative frequency of the class preceding the median class
- \( f \) `=` frequency of the median class
- \( w \) `=` width of the median class interval
This formula essentially interpolates within the median class interval to find the exact median value.
It's important to ensure that the data is properly grouped and that the cumulative frequency falls within the median class for this formula to be applicable.
Applications of Median
Median is a flexible statistical metric with many real-world uses in a wide range of industries. Here are a few typical uses for the median:
`1`. Income Distribution Analysis: When working with skewed data or outliers, median income is frequently used to reflect the normal income level within a community or group. It offers a more realistic portrayal of income distribution than mean income.
`2`. Real Estate Valuation: Particularly in places where there is a large disparity in property prices, the median sale price of properties within a particular area might give a better idea of typical property values than the mean.
`3`. Healthcare and Biostatistics: When data is not normally distributed or contains outliers, the median is commonly employed in medical research to characterise patient outcomes or features (e.g., age at diagnosis, time to recovery).
`4`. Demographic Analysis: By providing information about the average age of members of a group or community, the median age is frequently used to characterise the age distribution within a population.
`5`. Market research: By offering useful insights into consumer behaviour and market trends, median household expenditure or purchasing power can assist firms in making well-informed decisions about product pricing and marketing tactics.
Solved Examples
Example `1`. Find the median for the following data set : `10, 35, 25, 20, 30, 15`
Solution:
To find the median, follow these steps:
Step `1`: Arrange the data in ascending order:
\(10, 15, 20, 25, 30, 35\)
Step `2`: Since there are six observations (even), we need to find the average of the two middle values.
\(n = 6\)
Identify the position of the two middle values:
\( \text{Middle values} = \frac{n}{2} \text{ and } \frac{n}{2} + 1 \)
\( \text{Middle values} = \frac{6}{2} = 3 \text{ and } \frac{6}{2} + 1 = 4 \)
The third and fourth values are `20` and `25`.
Step `3`: Calculate the average of these middle values:
\( \text{Median} = \frac{20 + 25}{2} \)
\( \text{Median} = \frac{45}{2} \)
\( \text{Median} = 22.5 \)
So, the median of the given dataset is \( 22.5 \).
Example `2`. Find the median of the following dataset: `29, 23, 18, 12, 35`
Solution:
To find the median, follow these steps:
Step `1`: Arrange the data in ascending order:
\(12, 18, 23, 29, 35\)
Step `2`: Since there are five observations (odd), the median will be the middle value.
Identify the middle value:
\( \text{Middle value} = \frac{n + 1}{2} \)
\( \text{Middle value} = \frac{5 + 1}{2} = \frac{6}{2} = 3 \)
The third value in the ordered dataset is `23`.
Step `3`: So, the median of the given dataset is \( 23 \).
Example `3`. Find the median for the following grouped data:
Solution:
To find the median, follow these steps:
Step `1`: Calculate the cumulative frequency:
Step `2`: Determine the median class. Since the total number of observations is \(26\) (even), the median class is the class interval where the cumulative frequency exceeds \(\frac{n}{2} = \frac{26}{2} = 13\). So, the median class is \(21-30\).
Calculate the median using the formula:
\( M = L + \frac{\frac{n}{2} - F}{f} \times w \)
Where:
\(L\) `=` Lower boundary of the median class (in this case, \(L = 21\))
\(n\) `=` Total number of observations (\(n = 26\))
\(F\) `=` Cumulative frequency of the class before the median class (\(F = 9\))
\(f\) `=` Frequency of the median class (\(f = 8\))
\(w\) `=` Width of the median class interval (in this case, \(w = 10\))
\( M = 21 + \frac{\frac{26}{2} - 9}{8} \times 10 \)
\( M = 21 + \frac{13 - 9}{8} \times 10 \)
\( M = 21 + \frac{4}{8} \times 10 \)
\( M = 21 + 0.5 \times 10 \)
\( M = 21 + 5 \)
\( M = 26 \)
So, the median score of the class is \(26\).
Practice Problems
Q`1`. Find the median of the following ungrouped dataset:
\( 12, 7, 19, 5, 25, 38, 35 \)
- `19`
- `7`
- `12`
- `35`
Answer: a
Q`2`. Find the median of the following ungrouped dataset:
\( 8, 12, 16, 20, 24, 28, 32, 36, 40 \)
- `20`
- `25`
- `24`
- `40`
Answer: c
Q`3`. Find the median of the following grouped dataset:
- `43.25`
- `44.78`
- `40`
- `45.14`
Answer: d
Q`4`. Adrian’s aunt gifted him a piggy bank with `$20` for this birthday. Adrian added `$4` each week to the piggy bank and recorded the money he had in his piggy bank at the end of each week. If he did this for `7` weeks, what is the median of the amounts he recorded?
- `$36`
- `$40`
- `$32`
- `$44`
Answer: a
Q`5`. Find the median of the following grouped dataset:
- `15.75`
- `20.25`
- `23.5`
- `24.0`
Answer: c
Frequently Asked Questions
Q`1`. What is the median?
Answer: The median is a measure of central tendency that represents the middle value of a dataset when it's arranged in ascending or descending order. It divides the dataset into two equal halves, with half of the values falling below it and half above it.
Q`2`. How do you find the median of an ungrouped dataset?
Answer: To find the median of an ungrouped dataset, follow these steps:
`1`. Arrange the data in ascending or descending order.
`2`. If the number of observations (\(n\)) is odd, the median is the middle value. If \(n\) is even, the median is the average of the two middle values.
Q`3`. What if there are repeated values in the dataset?
Answer: When there are repeated values, treat each occurrence as a separate observation when determining the middle value. For example, in the dataset \(3, 4, 4, 5, 6\), the median would still be \(4\) because it's the middle value.
Q`4`. How do you find the median of a grouped dataset?
Answer: To find the median of a grouped dataset, you need to determine the median class, calculate the cumulative frequency, and then use the median formula for grouped data to interpolate within the median class interval and find the exact median value.
Q`5`. What is the significance of the median in statistics?
Answer: The median is significant because it's not influenced by extreme values or outliers in the dataset, unlike the mean. It provides a robust measure of central tendency, particularly in skewed distributions or datasets with outliers. Additionally, it's used in various statistical analyses and interpretations to understand the central tendency of the data.