Find the mode and median of 3, 4, 5, 5, 3, 6, 7, 3, 5, 5, 6.
A. 5 and 5
B. 3 and 5
C. 5 and 4
D. 3 and 4

Question

Find the mode and median of 3, 4, 5, 5, 3, 6, 7, 3, 5, 5, 6.A. 5 and 5B. 3 and 5C. 5 and 4D. 3 and 4

Prepp Team · Accepted Answer

Finding the Mode and Median of a Data Set

The question asks us to find two important measures of central tendency for the given set of numbers: the mode and the median.

The given data set is: 3, 4, 5, 5, 3, 6, 7, 3, 5, 5, 6.

Calculating the Mode

The mode is the number that appears most frequently in a data set. To find the mode, we can count how many times each number appears in the given set.

Let's list the numbers and their frequencies:

Number	Frequency (Count)
3	3
4	1
5	4
6	2
7	1

From the table, we can see that the number 5 appears 4 times, which is more than any other number in the set.

Therefore, the mode of the data set is 5.

Calculating the Median

The median is the middle value in a data set that is arranged in ascending or descending order. To find the median, we first need to arrange the given numbers in order.

Arranging the data set in ascending order:

3, 3, 3, 4, 5, 5, 5, 5, 6, 6, 7

Next, we need to find the total number of values in the data set. Counting the numbers, we find there are 11 values.

The total number of values, n, is 11.

Since the number of values (n=11) is odd, the median is the value at the \(\left(\frac{n+1}{2}\right)^{\text{th}}\) position in the ordered list.

Position of the median \( = \left(\frac{11+1}{2}\right)^{\text{th}} = \left(\frac{12}{2}\right)^{\text{th}} = 6^{\text{th}} \) position.

Now, let's look at the 6th number in the ordered list (3, 3, 3, 4, 5, 5, 5, 5, 6, 6, 7).

The number at the 6th position is 5.

Therefore, the median of the data set is 5.

Summary of Results

Mode = 5
Median = 5

We are looking for the option that gives 5 for the mode and 5 for the median.

Let's check the given options:

Option A: 5 and 5
Option B: 3 and 5
Option C: 5 and 4
Option D: 3 and 4

Our calculated mode is 5 and our calculated median is 5. This matches Option A.

Revision Table: Understanding Mode and Median

Measure	Definition	How to Calculate
Mode	The value that appears most frequently in a data set.	Count the occurrences 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.
Median	The middle value of a data set when it is ordered from least to greatest.	1. Arrange the data set in ascending or descending order. 2. If the number of values (n) is odd, the median is the value at the \( \left(\frac{n+1}{2}\right)^{\text{th}} \) position. 3. If the number of values (n) is even, the median is the average of the values at the \( \left(\frac{n}{2}\right)^{\text{th}} \) and \( \left(\frac{n}{2}+1\right)^{\text{th}} \) positions.

Additional Information: Measures of Central Tendency

Mode and median are types of central tendency measures. Another common measure is the mean (average).

Mean: Calculated by summing all the values in a data set and dividing by the total number of values. It represents the average value. The formula for the mean (\( \bar{x} \)) of a sample is \( \bar{x} = \frac{\sum x}{n} \), where \( \sum x \) is the sum of all values and \( n \) is the number of values.
When to use which measure:
- The mean is best for symmetrical data without extreme outliers.
- The median is good for skewed data or data with outliers because it is not affected by extreme values.
- The mode is useful for finding the most common category or value, especially for categorical or discrete data.

In this particular data set (3, 3, 3, 4, 5, 5, 5, 5, 6, 6, 7), the values are clustered around 5, and both the mode and median correctly reflect this central clustering.