The mean of the numbers 1, 4, 9, X, 12, 14, 15 and 16 is 10. Find the mode.
9
Finding the Mode of a Data Set Using the Mean
The problem asks us to find the mode of a set of numbers: 1, 4, 9, X, 12, 14, 15, and 16, given that their mean is 10. To find the mode, we first need to determine the value of X.
Step 1: Understand the Mean
The mean (or average) of a set of numbers is calculated by summing all the numbers in the set and dividing by the total count of numbers in the set.
The formula for the mean is:
$$ \text{Mean} = \frac{\text{Sum of all numbers}}{\text{Total count of numbers}} $$
Step 2: Calculate the Value of X
We are given the numbers 1, 4, 9, X, 12, 14, 15, and 16. There are a total of 8 numbers.
The given mean is 10.
Let's find the sum of the known numbers:
$$ \text{Sum of known numbers} = 1 + 4 + 9 + 12 + 14 + 15 + 16 $$
$$ \text{Sum of known numbers} = 71 $$
The sum of all numbers in the set is the sum of the known numbers plus X:
$$ \text{Sum of all numbers} = 71 + X $$
Now, we can plug this into the mean formula:
$$ 10 = \frac{71 + X}{8} $$
To solve for X, multiply both sides of the equation by 8:
$$ 10 \times 8 = 71 + X $$
$$ 80 = 71 + X $$
Subtract 71 from both sides:
$$ X = 80 - 71 $$
$$ X = 9 $$
So, the value of X is 9.
Step 3: Identify the Complete Set of Numbers
Now that we know X = 9, the complete set of numbers is:
$$ \{1, 4, 9, 9, 12, 14, 15, 16\} $$
Step 4: Find the Mode
The mode is the number that appears most frequently in a data set. Let's look at the frequency of each number in our complete set:
- 1 appears 1 time
- 4 appears 1 time
- 9 appears 2 times
- 12 appears 1 time
- 14 appears 1 time
- 15 appears 1 time
- 16 appears 1 time
The number that appears most often is 9, which occurs 2 times. All other numbers appear only once.
Therefore, the mode of the data set is 9.
| Number | Frequency |
|---|---|
| 1 | 1 |
| 4 | 1 |
| 9 | 2 |
| 12 | 1 |
| 14 | 1 |
| 15 | 1 |
| 16 | 1 |
Summary of Finding the Mode
We first used the given mean and the set of numbers with an unknown value X to calculate X. Once X was determined to be 9, we had the complete data set {1, 4, 9, 9, 12, 14, 15, 16}. By examining the frequency of each number, we found that 9 appeared most often, making it the mode.
Revision Table: Measures of Central Tendency
| Measure | Definition | How to Find | Use Case |
|---|---|---|---|
| Mean | The average value. | Sum of values divided by count. | Represents the typical value when data is symmetrical. |
| Median | The middle value. | Order data, find middle value (or average of two middle values). | Useful when data has outliers or is skewed. |
| Mode | The most frequent value. | Find value(s) with highest frequency. | Useful for categorical data or finding the most popular item. |
Additional Information on Statistics Concepts
Measures of central tendency like mean, median, and mode are fundamental concepts in statistics used to describe the center point of a data set. Each measure provides a different perspective on what constitutes a "typical" value.
- Mean: Sensitive to extreme values (outliers).
- Median: Resistant to extreme values, provides the exact middle point.
- Mode: Can be used for both numerical and categorical data. A data set can have one mode (unimodal), multiple modes (multimodal), or no mode if all values appear with the same frequency.
In this problem, finding the unknown value first was crucial to correctly identify the complete data set before calculating the mode.
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