Mean Median Mode
- Introduction
- Mean
- Median
- Mode
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
In Maths, mean, median, and mode are the three principal ways of finding the average value of any given data set. The mean, median, and mode are three measures of central tendency used to describe the typical or central value of a set of data. The mean is the average value of the data set, the median is the middle value when the data set is arranged in ascending order, and the mode is the value that appears most frequently in the data set. Let’s explore them in detail for various kinds of data sets.
Mean
When we add all the data points together and divide the result by the number of points, we get the mean of the data set. We call it arithmetic mean or average. It is denoted by `\bar{x}`.
Below is the formula to calculate mean:
There are two types of data whose mean can be found. Let’s discuss them:
Ungrouped Data
Suppose `x_1, x_2, x_3, x_4, .......,x_n` represents the `n` number of observations (data points). Below is the formula for calculating the mean of all the `n` data points.
Mean `= \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n}`
Example: What is the mean of the marks obtained by students in the mathematics test? The data (marks out of `50`) is given below with the names of students:
Solution: Sum of the marks obtained `= 45 + 50 + 35 + 25 + 30 = 185`
Number of students `= 5`
Mean `= (45 + 50 + 35 + 25 + 30)/5`
Mean `= 185/5 = 37`
Thus, the mean of the marks obtained by students in the mathematics test is `37`.
Grouped Data
Suppose `x_1, x_2, x_3, x_4, x_5, x_6,.......x_n` represent the `n` number of observations and `f_1, f_2, f_3, f_4, f_5, f_6,.......f_n` are their respective frequencies.
Below is the formula for calculating the mean of all the `n` number of observations.
Mean `= (x_1f_1+ x_2f_2+x_3f_3+...............+x_nf_n)/(f_1+f_2+ f_3+.............+f_n)` , or
Mean `= (∑x_nf_n)/(∑f_n)`, where `n = 1, 2, 3, …….`
Example: A dataset is given in tabular form. Determine the mean of the data.
Solution: First, we’ll find the product of the `x` and it’s corresponding `f` value.
Mean `= (∑x_nf_n)/(∑f_n) = 310/14 = 22.14`
Thus, the mean of the data given in the frequency table is `22.14`.
Median
If you arrange a set of data points in ascending order then the middle value of the arranged data is called the median.
There are two types of data whose median can be found. Let’s discuss them:
Ungrouped Data
For ungrouped data, we can come across two types of scenarios.
When the data set has an odd number of data points, we take the middle number as the median.
When the data set has an even number of data points, we take the average of the two middle numbers as median.
Grouped data
For grouped data, the data will be continuous and arranged in the frequency distribution. To find out the median of such type of data we can use the following formula.
Median `= L + [(n/2 - cf)/f] × h`
`L =` lower limit of the median class
`n =` number of observations
`cf =` cumulative frequency of class preceding the median class
`f =` frequency of the median class
`h =` class size
Mode
In a given data set, the number that appears the maximum number of times is called the mode.
Ungrouped data
For ungrouped data, the observation with the highest frequency is called mode.
Example: Find the mode of the given data points: `10, 20, 10, 30, 40, 60, 10, 40`.
Solution: In the given data set, `10` appears the maximum number of times in the set of data which means `10` is the mode of the given set of numbers.
Grouped data
For grouped data, if the data is continuous then there is a formula to calculate the mode.
Mode `= L + [(f_m - f_1)/(2f_m - f_1 - f_2)] × h`
`L =` lower limit of the modal class
`f_m =` frequency of modal class
`f_1 =` frequency of class preceding modal class
`f_2 =` frequency of class succeeding modal class
`h =` class size
Solved Examples
Example `1`. The time taken by Shelby to reach her school each day is listed below in the table. What is the mean time taken to reach school?
Solution:
As we know,
Mean `= "Sum of the data points"/"Number of data points"`
Mean `= (29 + 25 + 20 + 30 + 11)/5`
Mean `= 115/5 = 23`
Thus, the mean time taken by Shelby to reach school is `23` minutes.
Example `2`: A marketing company calculated the revenue earned per advertisement in `7` months and listed the data in a table given below. Determine the Median of the given data.
Solution:
To calculate the median first, we need to arrange the data into ascending order such as `11`, `20`, `25`, `29`, `30`, `31`, `34`
Now, we need to check whether the given data set has an even or an odd number of data points.
As we can observe from the given data the data is given only for `7` months and `7` is an odd number.
Therefore, the median is the middle number which is `29`.
Alternative method: Median = `((n + 1)/2)` th term
Median `= ((7 + 1)/2)` th term `= 4`th term `= 29`
Example `3`: Determine the median of the numbers: `10, 40, 50, 20, 30`.
Solution: First, we’ll arrange the numbers in ascending order such as `10`, `20`, `30`, `40`, `50`. Then we’ll look for the middle number, `((n + 1)/2)` th term, of the arranged numbers. The middle number will be our median.
Example `4`: Determine the median of the given grouped data.
Solution: First let’s calculate the cumulative frequency of the given grouped data.
The number of observations `n = 3 + 7 + 5 + 10 + 15 = 40`
So, `n/2 = 40/2 = 20`
Since `n` is even, we will find the average of the `(n/2)` th and the `((n + 1)/2)` th observation i.e. the cumulative frequency greater than `20` is `25`.
So the median class is class `18 -24`.
Median `= L + [(n/2 - cf)/f] × h`
This means, we need to choose the `f`, `cf`, and `h` from the class `(18-24)`
The lower limit of the class `(L)` `= 18`
Cumulative frequency `(cf)` of class preceding the median class `= 15`
Frequency `(f) = 10`,
Class size `(h) = 6` (since `24 - 18 = 6`)
Median `= 18 + ((20 - 15)/10)6`
Median `= 18 + 3`
Median `= 21`
Hence, the median of the given grouped data is `21`.
Example `5`: Determine the mode of the given set of data.
Solution:
Mode `= L + [(f_m - f_1)/(2f_m - f_1 - f_2)] × h`
The highest frequency `(f_m) = 14`
Therefore the modal class is `30 - 40`
Lower limit of the modal class `(L) = 30`
Preceding frequency `(f_1) = 9`
Succeeding frequency `(f_2) = 10`
Class size `(h) = 10`
Mode `= 30 + [(14 - 9)/(2(14) - 9 - 10)] × 10`
Mode `= 30 + 5/9 × 10`
Mode `= 30 + 5.56`
Mode `= 35.56`
Practice Problems
Q`1`. Determine the mean of the provided data rounded to the nearest whole number.
- `34`
- `35`
- `38`
- `36`
Answer: a
Q`2`. Determine the mode of the provided data, rounded to the first decimal place.
- `45`
- `44.5`
- `42`
- `43.5`
Answer: b
Q`3`. Determine the mode of the provided data.
`11, 20, 11, 30, 40, 60, 11, 40, 40, 40, 11, 30, 40, 11, 20, 11`
- `20`
- `30`
- `40`
- `11`
Answer: d
Q`4`. Determine the median of the provided data, rounded to the second decimal place.
- `47.57`
- `48.95`
- `43.26`
- `44.56`
Answer: a
Frequently Asked Questions
Q`1`. When is the mean the most appropriate measure of central tendency?
Answer: The mean is often used as a measure of central tendency when the data set is approximately symmetrically distributed and does not have extreme outliers. It is sensitive to extreme values and can be influenced by outliers.
Q`2`. When is the median the most appropriate measure of central tendency?
Answer: The median is typically used as a measure of central tendency when the data set has outliers or is skewed. It is less affected by extreme values compared to the mean and provides a more robust estimate of the central value in such cases.
Q`3`. When is the mode the most appropriate measure of central tendency?
Answer: The mode is useful as a measure of central tendency when identifying the most frequently occurring value is important, such as in categorical data or data with a clear peak or mode.
Q`4`. Can a data set have more than one mode?
Answer: Yes, a data set can have more than one mode. In cases where two or more values occur with the same highest frequency, the data set is said to be bimodal or multimodal.
Q`5`. What does it mean if the mean, median, and mode are equal?
Answer: If the mean, median, and mode are equal, it suggests that the data set is approximately symmetrically distributed and does not have any extreme outliers.
Q`6`. How are mean, median, and mode used in data analysis?
Answer: Mean, median, and mode are essential tools in data analysis for summarizing and understanding the central tendency of a data set. They provide valuable insights into the typical or central value of the data and help in making informed decisions in various fields, including statistics, economics, finance, and social sciences.