A collection of data’s center or average is described by central tendencies. Mean, median, and mode are the most often used markers of central tendency. In this article, we will learn to solve different problems based on mean, median and mode. This article also provides solution of mean, median and mode questions along with important formulas that need to solve them
Table of Content
- What is Mean?
- What is Median?
- What is Mode?
- Formulas of Mean, Median and Mode
- Solved Questions on Mean Median Mode
What is Mean, Median and Mode
- The mean of all the numbers in a dataset is its average. It is a central tendency measure. that is calculated by adding up all the values in a dataset and dividing by the total number of values.
- The median is the middle number in an ascending or descending dataset. If the data collection has odd numbers, the median is the middle number. If it has even values, the median is the middle two figures averaged.
- The value that shows up in a dataset the most often is called its mode. One mode, several modes, or none at all if no number repetitions can exist in a dataset.
The formulas of mean, median and mode are mentioned below:
Measure | Formula | Description |
|---|---|---|
Mean | Mean = [Tex]\frac{\sum \text{(all values in the dataset)}}{\text{Number of values in the dataset}}[/Tex] | Add all the values in the dataset and divide by the number of values. |
Median (Odd Numbers) | Median = [(n+1)/2]th value | Arrange the values in order and select the middle one. |
Median (Even Numbers) | Median= (n/2)th value and [(n/2) + 1]th value | Sort the values in ascending or descending order then average the two middle values. |
Median of grouped data | Median = [Tex]L + \left( \frac{\frac{N}{2} – CF}{f} \right) \times h[/Tex] | Where:
|
Mode | – | The value that appears most frequently in the dataset. |
Solved Questions on Mean Median Mode
Q1. Over ten days, a corporation has tracked the quantity of goods sold every day. The following statistics are provided:
Day | Number of Products Sold |
|---|---|
1 | 15 |
2 | 18 |
3 | 14 |
4 | 20 |
5 | 22 |
6 | 17 |
7 | 16 |
8 | 21 |
9 | 19 |
10 | 23 |
Over this ten-day period, figure out the average daily product sales.
Solution:
Sum of products sold= 15+18+14+20+22+17+16+21+19+23= 185
Number of days= 10
mean= 185/10
= 18.5
∴ The mean number of products sold per day is 18.5.
Q2. On a final exam, a lecturer noted each student’s score. This frequency distribution is the grouping of the scores:
Score Range | Frequency |
|---|---|
40-50 | 3 |
50-60 | 5 |
60-70 | 8 |
70-80 | 6 |
80-90 | 2 |
90-100 | 1 |
Calculate the mean score of the students.
Solution:
Score Range | Midpoint (x) | frequency (f) | f · x |
|---|---|---|---|
40-50 | 45 | 3 | 135 |
50-60 | 55 | 5 | 275 |
60-70 | 65 | 8 | 520 |
70-80 | 75 | 6 | 450 |
80-90 | 85 | 2 | 170 |
90-100 | 95 | 1 | 95 |
∑ (f · x) = 135+275+520+450+170+95
=1645
Total of frequency = ∑ (f)
∑ (f) = 3+5+8+6+2+1
=25
Mean = [Tex]\frac{\sum (f \cdot x)}{\sum f}[/Tex]
= 1645/25
= 65.8
∴ The mean score of the students is 65.8.
Q3. Find the median of the following dataset: 7,3,9,4,8
Solution:
Arrange the data in ascending order = 3,4,7,8,9
Number of numbers = 5 (odd)
Middle value = (n+1)/2
See AlsoWelche Rolle spielen Mittelwert, Median und Modus beim Vergleich verschiedener Datensätze?How do mean, median, and mode reflect the shape of a data distribution?= 6/2
=3
Median = 3rd Value of data set in arranged order.
Median = 7
Q4. Find the median of the following dataset: 12,5,8,10,14,7
Arrange the data in ascending order: 5,7,8,10,12,14
Number of values = 6 (even)
Middle values = 8 and 10
Median = [N1 + N2]/2
= [8 + 10] /2
= 18/2
= 9
∴ The median of the dataset is 9.
Q5. The following data shows the ages of participants in a workshop: 21,23,25,21,22,25,21,24,23,22,21,23,24,22,23
Solution:
Number | Frequency |
|---|---|
21 | 4 |
22 | 3 |
23 | 4 |
24 | 2 |
25 | 2 |
The numbers with the highest frequency= 21 and 23 (both appeared 4 times)
The dataset is bimodal, with modes 21 and 23.
Q6. Student homework hours were gathered through a survey. Following is a grouping of the results:
Hours Spent | Frequency |
|---|---|
0-5 | 4 |
6-10 | 8 |
11-15 | 12 |
16-20 | 6 |
21-25 | 2 |
Find the mean weekly homework hours spent.
Solution:
Hours Spent | Frequency (f) | Midpoint (x) | f· x |
|---|---|---|---|
0-5 | 4 | 2.5 | 10 |
6-10 | 8 | 8 | 64 |
11-15 | 12 | 13 | 156 |
16-20 | 6 | 18 | 108 |
21-25 | 2 | 23 | 46 |
Now,
∑ (f· x) = 10+64+156+108+46
= 384
Sum of frequencies = ∑f
= 4+8+12+6+2
= 32
Mean = ∑ (f· x) / ∑f
= 384/32
= 12
∴ The mean number of hours spent on homework per week is 12 hours.
Q7. The distribution of marks obtained by students in a class is given below
Marks range | Frequency |
|---|---|
0-20 | 4 |
21-40 | 6 |
41-60 | 10 |
61-80 | 8 |
81-100 | 2 |
Solution:
Marks range | Frequency | Cumulative Frequency (CF) |
|---|---|---|
0-20 | 4 | 4 |
21-40 | 6 | 10 |
41-60 | 10 | 20 |
61-80 | 8 | 28 |
81-100 | 2 | 30 |
In this case the formula to find the median will be :[Tex]L + \left( \frac{\frac{N}{2} – CF}{f} \right)[/Tex]
We have, total number of observations (N) = 30
N/2 = 30/2 = 15
The Median Class = is the class where the cumulative frequency just exceeds 1515. Here, the median class is 41−6041−60 since its cumulative frequency (20) exceeds 15.
- L=40.5 (Lower boundary of the median class)
- N=30 (Total number of observations)
- CF=10 (Cumulative frequency of the class preceding the median class)
- f=10 (Frequency of the median class)
- h = 20 (class width)
Putting these values in the formula we get,
Median = [Tex]40.5 + \left( \frac{15 – 10}{10} \right) \times 20[/Tex]
= [Tex]40.5 + \left( \frac{5}{10} \right) \times 20[/Tex]
= 40.5+0.5×20
= 40.5 + 10
= 50.5
Thus, the median marks are 50.5.
Q8. The table below shows the number of times different types of food were chosen by customers in a week:
Food | Frequency |
|---|---|
Burger | 14 |
Pizza | 20 |
Sandwich | 12 |
Salad | 10 |
Pasta | 18 |
Solution:
From the frequencies given, the maximum number of times any food occurred is pizza.
Therefore, Pizza is the mode of choice
1. The ages of a group of friends are: 25, 30, 28, 32, and 35. Find the mean age of the group.
2. The test scores of a class are: 65, 72, 78, 85, 90. Find the median score.
3. A football team played 10 matches. The number of goals scored by the team are: 2, 3, 1, 4, 2, 3, 2, 5, 4, 2. Find the mode.
4. The weights of five students in a class are: 50 kg, 55 kg, 60 kg, 65 kg, and 70 kg. Find the mean weight.
5. The number of siblings of students in a school are: 0, 1, 2, 1, 3, 2, 1, 2, 1, 0. Find the mode.
6. The heights of students in a class are: 150 cm, 155 cm, 160 cm, 165 cm, and 170 cm. Find the median height.
7. The monthly salaries (in $) of employees in a company are: 3000, 3500, 4000, 4500, and 5000. Find the mean salary.
8. The ages of a group of people are: 25, 30, 35, 40, 45, 50, 55. Find the median age.
9. The temperatures (in °C) recorded over a week are: 20, 22, 25, 24, 26, 23, 21. Find the mean temperature.
10. The number of hours spent studying by students in a week are: 2, 3, 4, 2, 5, 3, 2, 4, 3, 3. Find the mode.
FAQs on Mean Median Mode
What is the relation between mean, median and mode?
The mean, median, and mode are measures of central tendency used to summarize a set of data. The relation between mean, median and mode can be given by: Mode = 3 Median – 2 Mean
How to calculate mean?
To calculate the mean (average) of a set of numbers:
- Sum all the values: Add up all the numbers in the dataset.
- Divide by the number of values: Count the total number of values in the dataset and divide the sum by this count.
The formula = Σn / N
How to calculate median?
The median is the midpoint of the dataset, when arranged in an order.
- For odd numbers median = (n + 1)/2 th term
- For even numbers median = [(n/2)th term + {(n/2) + 1}th term]/2
What is Mode of a Data?
The data set occurring most number of times is mode of the given data
What is Frequency of a dataset?
The frequency of a dataset tells how many times a particular data appears or occurs
shreyaagrawal65
Improve
Previous Article
PAT3_SQL | Question 4
Next Article
Riemann Zeta Function
