Question
of dataset 40,38,62,70,56,41.58,48,60,45
Answers
GPT-4o mini
To analyze the dataset you provided (40, 38, 62, 70, 56, 41, 58, 48, 60, 45), we can calculate several statistical measures, including the mean, median, mode, range, variance, and standard deviation.
### 1. Mean
The mean (average) is calculated by summing all the numbers and dividing by the count of numbers.
\[
\text{Mean} = \frac{40 + 38 + 62 + 70 + 56 + 41 + 58 + 48 + 60 + 45}{10} = \frac{468}{10} = 46.8
\]
### 2. Median
To find the median, we first need to sort the dataset in ascending order:
\[
38, 40, 41, 45, 48, 56, 58, 60, 62, 70
\]
Since there are 10 numbers (an even count), the median is the average of the 5th and 6th numbers:
\[
\text{Median} = \frac{48 + 56}{2} = \frac{104}{2} = 52
\]
### 3. Mode
The mode is the number that appears most often. In this dataset, all numbers appear only once, so there is no mode.
### 4. Range
The range is calculated by subtracting the smallest number from the largest number:
\[
\text{Range} = 70 - 38 = 32
\]
### 5. Variance
Variance measures how spread out the numbers are. It is calculated as follows:
1. Find the mean (calculated previously as 46.8).
2. Subtract the mean from each number, square the result, and then find the average of those squared differences.
\[
\text{Variance} = \frac{(40 - 46.8)^2 + (38 - 46.8)^2 + (62 - 46.8)^2 + (70 - 46.8)^2 + (56 - 46.8)^2 + (41 - 46.8)^2 + (58 - 46.8)^2 + (48 - 46.8)^2 + (60 - 46.8)^2 + (45 - 46.8)^2}{10}
\]
Calculating each term:
- \( (40 - 46.8)^2 = 46.24 \)
- \( (38 - 46.8)^2 = 77.44 \)
- \( (62 - 46.8)^2 = 231.84 \)
- \( (70 - 46.8)^2 = 534.24 \)
- \( (56 - 46.8)^2 = 83.64 \)
- \( (41 - 46.8)^2 = 33.64 \)
- \( (58 - 46.8)^2 = 124.96 \)
- \( (48 - 46.8)^2 = 1.44 \)
- \( (60 - 46.8)^2 = 174.24 \)
- \( (45 - 46.8)^2 = 3.24 \)
Sum of squared differences:
\[
46.24 + 77.44 + 231.84 + 534.24 + 83.64 + 33.64 + 124.96 + 1.44 + 174.24 + 3.24 = 1097.6
\]
Now, divide by the number of data points (10):
\[
\text{Variance} = \frac{1097.6}{10} = 109.76
\]
### 6. Standard Deviation
The standard deviation is the square root of the variance:
\[
\text{Standard Deviation} = \sqrt{109.76} \approx 10.48
\]
### Summary
- **Mean**: 46.8
- **Median**: 52
- **Mode**: No mode
- **Range**: 32
- **Variance**: 109.76
- **Standard Deviation**: 10.48
Feel free to ask if you need more analysis or information!
### 1. Mean
The mean (average) is calculated by summing all the numbers and dividing by the count of numbers.
\[
\text{Mean} = \frac{40 + 38 + 62 + 70 + 56 + 41 + 58 + 48 + 60 + 45}{10} = \frac{468}{10} = 46.8
\]
### 2. Median
To find the median, we first need to sort the dataset in ascending order:
\[
38, 40, 41, 45, 48, 56, 58, 60, 62, 70
\]
Since there are 10 numbers (an even count), the median is the average of the 5th and 6th numbers:
\[
\text{Median} = \frac{48 + 56}{2} = \frac{104}{2} = 52
\]
### 3. Mode
The mode is the number that appears most often. In this dataset, all numbers appear only once, so there is no mode.
### 4. Range
The range is calculated by subtracting the smallest number from the largest number:
\[
\text{Range} = 70 - 38 = 32
\]
### 5. Variance
Variance measures how spread out the numbers are. It is calculated as follows:
1. Find the mean (calculated previously as 46.8).
2. Subtract the mean from each number, square the result, and then find the average of those squared differences.
\[
\text{Variance} = \frac{(40 - 46.8)^2 + (38 - 46.8)^2 + (62 - 46.8)^2 + (70 - 46.8)^2 + (56 - 46.8)^2 + (41 - 46.8)^2 + (58 - 46.8)^2 + (48 - 46.8)^2 + (60 - 46.8)^2 + (45 - 46.8)^2}{10}
\]
Calculating each term:
- \( (40 - 46.8)^2 = 46.24 \)
- \( (38 - 46.8)^2 = 77.44 \)
- \( (62 - 46.8)^2 = 231.84 \)
- \( (70 - 46.8)^2 = 534.24 \)
- \( (56 - 46.8)^2 = 83.64 \)
- \( (41 - 46.8)^2 = 33.64 \)
- \( (58 - 46.8)^2 = 124.96 \)
- \( (48 - 46.8)^2 = 1.44 \)
- \( (60 - 46.8)^2 = 174.24 \)
- \( (45 - 46.8)^2 = 3.24 \)
Sum of squared differences:
\[
46.24 + 77.44 + 231.84 + 534.24 + 83.64 + 33.64 + 124.96 + 1.44 + 174.24 + 3.24 = 1097.6
\]
Now, divide by the number of data points (10):
\[
\text{Variance} = \frac{1097.6}{10} = 109.76
\]
### 6. Standard Deviation
The standard deviation is the square root of the variance:
\[
\text{Standard Deviation} = \sqrt{109.76} \approx 10.48
\]
### Summary
- **Mean**: 46.8
- **Median**: 52
- **Mode**: No mode
- **Range**: 32
- **Variance**: 109.76
- **Standard Deviation**: 10.48
Feel free to ask if you need more analysis or information!