Instructions for Computing the Median and Real-World Examples

The median, a statistical measure of central tendency, is the value of the middle data point. Using appropriate formulas, we can determine the median of various data sets, including both ungrouped and grouped data. This article will teach you instructions for determining the middle value of given data set with extensive explanations and worked-out examples

Let's review the differences between grouped and ungrouped information.

Summarized information constitutes information that has been sorted into classes after being compiled. Using a frequency table, the raw data can be organized into distinct categories.

Unclassified information refers to information that has not been categorized or organized in any way since it was gathered. Since data typically consists of numerical or descriptive information, ungrouped data is information that has not been organized in this way.

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Median

The median is centered among a set of numbers that have been sorted from smallest to largest When there are an even number of values, the median is found by averaging the middle two.

Guidelines for determining the middle value in a data set

• Order the figures from largest to smallest.
• When there are an odd number of terms, the median value is the one that falls in the middle.
• If there are an even number of terms, you can simply add the middle two together and divide by 2.

Click here to read up on data medians.

Finding the Median in a Group of Random Numbers

We can determine the median of numbers without grouping them if we count them. So, let's find out how to calculate the median of a data set.

If the count of observations is unequal to 2, 3, etc.

Prior to doing anything else, sort the data by ascending or descending order. Determine how many samples are included in the current data set.

N (an odd number) is the total number of data points.

The median is defined as the middle value in a set of data:

Read the following case study to learn how to calculate the median.

Example 1 During their winter break, the Clintons drove through seven states, each with its own unique gas price. Determine the midpoint of these gas prices

$1 59, $1 31, $1 96, $3 09, $1 64, $1 55, $2 61

Solution : 

The following expression provides an ascending order for the provided data:

$1 31, $1 55, $1 59, $1 64, $1 96, $2 61, $3 09

In this case, there are seven possible values.

The median is defined as the halfway point between any two observations (n = 1).

It's the =[(7 1)/2]th time that's been observed

The Fourth Observation =

= $1 64

That's why $1 is the current average price for a gallon of gas. 64

If the total is an even number:

After sorting the data in either ascending or descending order, count the number of observations, i. e n Given that, if n is an even number, then

Median = Average of the [(n2/1)th] and [(n2/1)th] Observation

Take a look at the worked example below:

Example 2: In a Kabaddi tournament, a team's point total looks like this after each game:

17, 2, 7, 27, 15, 5, 14, 8, 10, 24, 48, 10, 8, 7, 18, 28 

Find the middle score the team made.  

Solution: 

By sorting the team's point totals from highest to lowest, we get;

2, 5, 7, 7, 8, 8, 10, 10, 14, 15, 17, 18, 24, 27, 28, 4

Data from 16 separate experiments.

n/2 = 16/2 = 8

(n/2) 1 = (16/2) 1 = 8 1 = 9

Eighth and ninth data points will be averaged to determine the median.

The median is equal to 12 (10 + 14)/2 = 24

Therefore, 12 is the middle point of the Kabaddi team's point total.

Find the middle number in a set of numbers by remembering what the cumulative frequency was before learning how to find the median.

Total occurrences allows one to determine the fraction of data points that fall above (or below) a given threshold Finding the total frequency requires consulting a frequency distribution table. Cumulative frequency is calculated by adding each successive frequency from a frequency distribution table to the sum of the frequencies that came before it. Given that every frequency is already factored into the previous total, the final value will always be equal to the total number of observations.

Finding the Middle Class in a List of Items

Find the value of N2 if N is the total number of frequencies.

The median class of grouped data is the group whose cumulative frequency is most frequently between the two extremes, N/2 and N.

Understanding the Methods for Determining the Median of a Set of Data

For a given frequency distribution, we can use the following formula to determine the middle value, i. e , for data that has been organized into groups, is:

To calculate the median, we use the formula: Median = l [(N/2 - cf)/f] h

Here,

The lower limit of the middle class is denoted by the symbol l.

N = Accumulated Occurrences

cf = Frequency of classes above and below the median

The frequency of the middle class (f = )

height of the class = h

In order to see how to use this formula to extract the median from a frequency table, please review the problem that has been solved below.

Question: Find the median of the following data set.

Solution:

In order to compute the cumulative frequency of the given frequency distribution, we must first transform the discontinuous classes into continuous ones.

N = 36

N/2 = 36/2 = 18

27 is the cumulative frequency over and near 18, which is in the class interval 54. 5 – 59 5

In this case, the middle class is equal to 54. 5 – 59 5

class l = minimum of the middle fifty-four 5

The number of observations, N, is equal to the sum of 36 frequencies.

Cumulative frequency of the class before the median, expressed as cf = 15

Median Class Frequency = f = 12

In this case, h = Class Height = 5.

The median is calculated as follows:

= 54 5 [(18 – 15)/12] × 5

= 54 5 [(3 × 5)/12]

= 54 4 (5/4)

= 54 5 1 25

= 55 75

With this information, we can conclude that the median of the given distribution is 55. 75

Discovering the Middle of a Set of Numbers that Can Be Divided Into Two Groups

If each observation in a data set can be treated as independent of every other observation in the set, then the set is said to be discrete. When the data is organized into discrete sets of observations of equal size, the median can be calculated as the ((n 1)/2)th observation. The median is the value in the cumulative frequency distribution at which the ((n 1)/2)th observation falls.

Example:

The number of cabs available for use by various office locations is used to categorize the given frequency distribution. Determine the mean number of taxis in operation.

Solution:

The median is defined as the value that represents the middle value of a set of data.

= (101 1)/2

= 102/2

Here we present our fifty-first remark:

The median is 7, since that's what the middle number is.

Here's a video that will teach you how to calculate the median in any data set.

As of right now, we have been able to determine the data's middle point using a variety of methods and formulas. Let's check out the graphical method for determining the middle value.

Visual Methods for Determining the Median

When we draw Ogive curves that are less than and more than the given distribution, we can easily determine the median. Here's an illustration that should help clarify the point:

If 30 stores in a shopping mall in a given area each had a $10,000 profit last year, the profits would be distributed as follows.

To find the middle point, just draw the ogives that are less than and more than the one you're interested in.

Solution:

You can make the following tables with the provided information:

First, we sketch the coordinate axes, with the cumulative frequency going up the vertical axis and the minimum possible profit going across the horizontal.

Next, for all Ogives other than type I, we plot the coordinates (5, 30), (10, 28), (15, 16), (20, 14), (25, 10), (30, 7), and (35, 3).

If you plot the upper limits of class intervals and the corresponding cumulative frequency, like (10, 2), (15, 14), (20, 16), (25, 20), (30, 23), (35, 27), and (40, 30) on the same axes, you'll get the less than type Ogive.

However, when we plot all these points, we find that the abscissa of their intersection is close to 17. As can be seen in the following graph, the middle value is 5.

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