If the mean of the numbers 2, (p + 1), 8, 19, 16, 3 and p is 9, then find the mode of the numbers.
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Understanding how to calculate the mean and mode of a set of numbers is fundamental in statistics. This problem requires us to first use the given mean to find a missing value in the data set and then determine the mode of the completed set.
Calculating the Missing Value Using the Mean
The mean of a set of numbers is the sum of all the numbers divided by the total count of numbers. We are given the numbers 2, (p + 1), 8, 19, 16, 3, and p, and their mean is 9. There are 7 numbers in total.
Let's write down the given information:
- Numbers: 2, (p + 1), 8, 19, 16, 3, p
- Count of numbers: 7
- Mean: 9
The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of numbers}}{\text{Count of numbers}} \]First, let's find the sum of the given numbers:
\[ \text{Sum} = 2 + (p + 1) + 8 + 19 + 16 + 3 + p \] \[ \text{Sum} = 2 + p + 1 + 8 + 19 + 16 + 3 + p \]Combine the constant terms and the terms with p:
\[ \text{Sum} = (2 + 1 + 8 + 19 + 16 + 3) + (p + p) \] \[ \text{Sum} = 49 + 2p \]Now, we can use the mean formula to set up an equation and solve for p:
\[ 9 = \frac{49 + 2p}{7} \]Multiply both sides of the equation by 7:
\[ 9 \times 7 = 49 + 2p \] \[ 63 = 49 + 2p \]Subtract 49 from both sides:
\[ 63 - 49 = 2p \] \[ 14 = 2p \]Divide both sides by 2:
\[ p = \frac{14}{2} \] \[ p = 7 \]So, the value of p is 7.
Finding the Mode of the Numbers
Now that we know p = 7, we can list the complete set of numbers:
- 2
- (p + 1) = (7 + 1) = 8
- 8
- 19
- 16
- 3
- p = 7
The set of numbers is 2, 8, 8, 19, 16, 3, 7.
The mode of a set of numbers is the number that appears most frequently in the set. To find the mode, we need to count the frequency of each number in the set {2, 8, 8, 19, 16, 3, 7}.
| Number | Frequency (Count) |
|---|---|
| 2 | 1 |
| 8 | 2 |
| 19 | 1 |
| 16 | 1 |
| 3 | 1 |
| 7 | 1 |
From the table, we can see that the number 8 appears 2 times, which is more than any other number in the set. Therefore, the mode of the numbers is 8.
Conclusion
By using the given mean, we first found the value of the unknown variable p. Once p was determined, we were able to list all the numbers in the set and then identify the number that occurred most frequently, which is the mode.
Revision Table: Mean and Mode
| Concept | Definition | How to Calculate |
|---|---|---|
| Mean | The average of a set of numbers. | Sum of all numbers divided by the count of numbers. |
| Mode | The number that appears most frequently in a data set. | Count the occurrences of each number; the number with the highest frequency is the mode. A data set can have one mode (unimodal), multiple modes (multimodal), or no mode. |
Additional Information: Measures of Central Tendency
Mean and mode are types of averages, known as measures of central tendency. They help summarize a data set by giving a single value that represents the center or typical value of the data.
- Mean: Best used for symmetrical data sets without outliers.
- Mode: Useful for finding the most common item or category, especially in non-numerical data. It is the only measure of central tendency suitable for nominal data.
- Median: The middle value in a data set that is ordered from least to greatest. It is less affected by outliers than the mean and is useful for skewed data.
Choosing the right measure of central tendency depends on the type of data and the distribution of the data.
Important Questions from Average
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In a particular week, the average earning per day of a plumber from Monday to Wednesday remained ₹580 and from Thursday to Saturday it was ₹612. If the average earning for the whole week was ₹675, then how much did he earn on Sunday?
