What is the mode of 21, 22, 22, 23, 24, 24, 24?
24
Finding the Mode of a Data Set
The question asks us to find the mode of the given set of numbers: 21, 22, 22, 23, 24, 24, 24.
The mode is a fundamental concept in statistics and represents one of the measures of central tendency. It is defined as the value that appears most frequently in a data set.
To find the mode, we need to count how many times each number appears in the given set. Let's list the numbers and their frequencies:
- 21 appears 1 time.
- 22 appears 2 times.
- 23 appears 1 time.
- 24 appears 3 times.
We can also represent this frequency count in a table:
| Number | Frequency (How many times it appears) |
|---|---|
| 21 | 1 |
| 22 | 2 |
| 23 | 1 |
| 24 | 3 |
Comparing the frequencies, we see that:
- The number 21 has a frequency of 1.
- The number 22 has a frequency of 2.
- The number 23 has a frequency of 1.
- The number 24 has a frequency of 3.
The number with the highest frequency is 24, which appears 3 times. Therefore, the mode of the data set {21, 22, 22, 23, 24, 24, 24} is 24.
In mathematical terms, the mode is the value \(x_i\) with the highest frequency \(f_i\). In this case, the highest frequency is 3, and the corresponding value is 24.
Thus, the mode is 24.
Revision Table: Measures of Central Tendency
Understanding the mode is part of understanding measures of central tendency. Here's a quick overview of the three main measures:
| Measure | Definition | How to Calculate | Use Case |
|---|---|---|---|
| Mean (Average) | The sum of all values divided by the number of values. | Sum all values, then divide by the count of values. | Used for symmetrical data sets, when you want the average value. |
| Median (Middle Value) | The middle value of a data set when ordered from least to greatest. | Order the data. If the count is odd, it's the middle number. If even, it's the average of the two middle numbers. | Used for skewed data sets or data with outliers, as it's not affected by extreme values. |
| Mode (Most Frequent) | The value that appears most often in a data set. | Count the frequency of each value. The mode is the value with the highest frequency. A set can have one mode (unimodal), multiple modes (multimodal), or no mode. | Used for categorical data or when you want to know the most common item or value. |
Additional Information: Understanding Mode in Statistics
The mode is a useful measure especially when dealing with non-numeric data or when trying to find the most popular item in a category. For example, if you ask students their favorite color, the mode would tell you which color is the most liked.
Unlike the mean and median, a data set can have more than one mode. If two or more values have the same highest frequency, then all of those values are considered modes. This is called a multimodal data set.
Conversely, if all values in a data set appear the same number of times (e.g., each value appears only once), the data set is said to have no mode.
In our problem, only one value (24) appeared with the highest frequency (3), so the data set is unimodal.
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