A, B, C are three sets of values of x: A: 2, 3, 7, 1, 3, 2, 3 B: 7, 5, 9, 12, 5 3, 8 C: 4, 4, 11, 7, 2, 3, 4 Select the correct statement from among the following
Mean, Median and Mode of A are same
Understanding Measures of Central Tendency for Data Sets
This question requires us to calculate the mean, median, and mode for three different sets of numerical data (Sets A, B, and C) and then evaluate several statements comparing these measures across the sets.
Let's break down the process for each set.
Calculating Mean, Median, and Mode for Set A
Set A: 2, 3, 7, 1, 3, 2, 3
First, let's arrange the values in ascending order to easily find the median:
- Ordered Set A: 1, 2, 2, 3, 3, 3, 7
The number of values in Set A (n) is 7.
Mean of Set A:
The mean is the sum of all values divided by the number of values.
Formula for Mean ($\bar{x}$): $\bar{x} = \frac{\sum x}{n}$
Sum of values in A: $1 + 2 + 2 + 3 + 3 + 3 + 7 = 21$
Mean of A: $\frac{21}{7} = 3$
Median of Set A:
The median is the middle value of an ordered data set. Since n = 7 is an odd number, the median is the value at the $\frac{n+1}{2}$ position.
Position of Median: $\frac{7+1}{2} = \frac{8}{2} = 4\text{th position}$
In the ordered Set A (1, 2, 2, 3, 3, 3, 7), the 4th value is 3.
Median of A: 3
Mode of Set A:
The mode is the value that appears most frequently in the set.
In Set A (2, 3, 7, 1, 3, 2, 3), the value 3 appears 3 times, which is more than any other value.
Mode of A: 3
Calculating Mean, Median, and Mode for Set B
Set B: 7, 5, 9, 12, 5, 3, 8
Arrange in ascending order:
- Ordered Set B: 3, 5, 5, 7, 8, 9, 12
The number of values in Set B (n) is 7.
Mean of Set B:
Sum of values in B: $3 + 5 + 5 + 7 + 8 + 9 + 12 = 49$
Mean of B: $\frac{49}{7} = 7$
Median of Set B:
Position of Median: $\frac{7+1}{2} = 4\text{th position}$
In the ordered Set B (3, 5, 5, 7, 8, 9, 12), the 4th value is 7.
Median of B: 7
Mode of Set B:
In Set B (7, 5, 9, 12, 5, 3, 8), the value 5 appears 2 times, which is more than any other value.
Mode of B: 5
Calculating Mean, Median, and Mode for Set C
Set C: 4, 4, 11, 7, 2, 3, 4
Arrange in ascending order:
- Ordered Set C: 2, 3, 4, 4, 4, 7, 11
The number of values in Set C (n) is 7.
Mean of Set C:
Sum of values in C: $2 + 3 + 4 + 4 + 4 + 7 + 11 = 35$
Mean of C: $\frac{35}{7} = 5$
Median of Set C:
Position of Median: $\frac{7+1}{2} = 4\text{th position}$
In the ordered Set C (2, 3, 4, 4, 4, 7, 11), the 4th value is 4.
Median of C: 4
Mode of Set C:
In Set C (4, 4, 11, 7, 2, 3, 4), the value 4 appears 3 times, which is more than any other value.
Mode of C: 4
Summary of Calculated Measures
| Set | Mean | Median | Mode |
|---|---|---|---|
| A | 3 | 3 | 3 |
| B | 7 | 7 | 5 |
| C | 5 | 4 | 4 |
Evaluating the Given Statements
Let's check each statement based on our calculations:
- Mean, Median and Mode of A are same: Mean of A = 3, Median of A = 3, Mode of A = 3. All three measures are equal to 3. This statement is True.
- Mean of C is equal to Median of B: Mean of C = 5, Median of B = 7. $5 \neq 7$. This statement is False.
- Median of B is equal to Mode of A: Median of B = 7, Mode of A = 3. $7 \neq 3$. This statement is False.
- Mean of A is equal to Mode of C: Mean of A = 3, Mode of C = 4. $3 \neq 4$. This statement is False.
Based on our evaluation, only the first statement is correct.
Revision Table: Key Statistics Concepts
| Measure | Definition | How to Calculate (for ungrouped data) |
|---|---|---|
| Mean | The average of the data set. | Sum of all values divided by the number of values. $\left( \bar{x} = \frac{\sum x}{n} \right)$ |
| Median | The middle value of an ordered data set. | Arrange data in order. For odd $n$, it's the value at the $\frac{n+1}{2}$ position. For even $n$, it's the average of the values at the $\frac{n}{2}$ and $\frac{n}{2}+1$ positions. |
| Mode | The value that appears most frequently. | Count the frequency of each value. The value with the highest frequency is the mode. A data set can have one mode (unimodal), multiple modes (multimodal), or no mode. |
Additional Information: Central Tendency Measures
Mean, Median, and Mode are called measures of central tendency because they describe the center point of a data set. Each measure has its strengths and weaknesses and provides a different perspective on the typical value in the data.
- The Mean is sensitive to extreme values (outliers).
- The Median is not affected by extreme values, making it a good measure for skewed data.
- The Mode is useful for finding the most common item or category, and it is the only measure of central tendency that can be used with nominal data.
Understanding these measures helps in summarizing and interpreting data sets effectively in statistics.
Important Questions from Measures of Central Tendency
A random sample of 20 people is classified in the following table according to their ages:
Age
Frequency
15 – 25
2
25 – 35
4
35 – 45
6
45 – 55
5
55 - 65
3
What is the mean age of this group of people?
The median of the following observations 46, 64, 87, 41, 58, 77, 35, 90, 55, 92, 33 is 58. If 92 is replaced by 99 and 41 by 43 in the above data. The new median is:
If the difference of mode and median is 36, then the difference of median and mean is:
In a Mathematics test 15 students scored 80 marks, 20 students scored 75 marks, 28 students scored 65 marks and 25 students scored 60 marks, mode of the score is:
If the mode of the scores 10, 12, 13, 15, 15, 13, 12, 10, x is 15, then what is the value of x?
