Sakshi attended to the following number of clients at the front desk during her internship for 15 days: 18, 20, 16, 17, 32, 17, 6, 16, 12, 13, 17, 28, 24, 45, 17. Find the average of the mode and median of the given data.
17
Understanding Data Analysis: Mode and Median
The question asks us to calculate the average of the mode and the median for a given set of data representing the number of clients Sakshi attended to over 15 days.
The given data is: 18, 20, 16, 17, 32, 17, 6, 16, 12, 13, 17, 28, 24, 45, 17.
To find the average of the mode and median, we first need to determine the mode and the median of this dataset.
Finding the Mode of the Data
The mode is the value that appears most frequently in a dataset. To find the mode, we can count the occurrences of each number in the given data.
Let's list the numbers and their frequencies:
- 6: 1 time
- 12: 1 time
- 13: 1 time
- 16: 2 times
- 17: 4 times
- 18: 1 time
- 20: 1 time
- 24: 1 time
- 28: 1 time
- 32: 1 time
- 45: 1 time
The number that appears most often is 17, which occurs 4 times.
Therefore, the mode of the data is 17.
Finding the Median of the Data
The median is the middle value in a dataset that is ordered from least to greatest. To find the median, we must first arrange the data in ascending order.
The given data is: 18, 20, 16, 17, 32, 17, 6, 16, 12, 13, 17, 28, 24, 45, 17.
Arranging the data in ascending order:
6, 12, 13, 16, 16, 17, 17, 17, 17, 18, 20, 24, 28, 32, 45
There are 15 data points. Since there is an odd number of data points (n = 15), the median is the value at the position $\frac{n+1}{2}$.
Position of the median = $\frac{15+1}{2} = \frac{16}{2} = 8\text{th}$ position.
Let's find the 8th value in the ordered list:
| Position | Value |
|---|---|
| 1st | 6 |
| 2nd | 12 |
| 3rd | 13 |
| 4th | 16 |
| 5th | 16 |
| 6th | 17 |
| 7th | 17 |
| 8th (Median) | 17 |
| 9th | 17 |
| 10th | 18 |
| 11th | 20 |
| 12th | 24 |
| 13th | 28 |
| 14th | 32 |
| 15th | 45 |
The value at the 8th position is 17.
Therefore, the median of the data is 17.
Calculating the Average of Mode and Median
Now that we have found the mode and the median, we can calculate their average.
Mode = 17
Median = 17
Average = $\frac{\text{Mode} + \text{Median}}{2}$
Average = $\frac{17 + 17}{2}$
Average = $\frac{34}{2}$
Average = 17
The average of the mode and median is 17.
Final Answer Summary
Given data: 18, 20, 16, 17, 32, 17, 6, 16, 12, 13, 17, 28, 24, 45, 17
- Mode = 17 (most frequent value)
- Median = 17 (middle value in ordered data)
- Average of Mode and Median = $\frac{17 + 17}{2} = 17$
Revision Table: Key Statistical Concepts
| Concept | Definition | How to Find |
|---|---|---|
| Mode | The value that appears most frequently in a dataset. A dataset can have one mode (unimodal), multiple modes (multimodal), or no mode. | Count the frequency of each value. The value with the highest frequency is the mode. |
| Median | The middle value in a dataset when the data is arranged in ascending or descending order. It divides the dataset into two equal halves. | 1. Order the data. 2. If $n$ (number of data points) is odd, the median is the value at the $\frac{n+1}{2}$ position. 3. If $n$ is even, the median is the average of the values at the $\frac{n}{2}$ and $\frac{n}{2}+1$ positions. |
| Mean (Average) | The sum of all values in a dataset divided by the number of values. | Sum all values and divide by the total count of values. Formula: $\text{Mean} = \frac{\sum x}{n}$ |
Additional Information: Measures of Central Tendency
Mode, median, and mean are all measures of central tendency. They are single values that attempt to describe a set of data by identifying the central position within that set of data.
- The mean is sensitive to extreme values (outliers).
- The median is less affected by extreme values and is a good measure for skewed distributions.
- The mode is useful for categorical data or for identifying the most common value in a dataset. In this specific problem, the mode and median happened to be the same value, which is interesting but not always the case. Understanding how to calculate each measure independently is key.
Important Questions from Elementary Statistics
In a colony 5 families have 1 child, 7 families have 2 children, 8 families have 3 children and 3 families have 4 children.What is the mode of the number of children.
What will be the difference between mean and median of the given data?
21, 11, 27, 8, 5, 12, 7, 23, 3, 14, 9, 19Find the median of the data 10, 8, 5, 3, 9, 6, 12, 14, 13, 7, 1
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
For which set of numbers do the mean, median and mode all have the same value?
