The standard deviation is a measure of how spread out the numbers in a data set are. It gives an idea of how much individual values deviate from the mean (average) of the data set.

To calculate the standard deviation, follow these steps:

Step 1: Find the Mean

First, calculate the mean of the data set.

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$

Given the values: $25, 40, 50, 50, 50, 50, 50, 50, 60, 60, 60, 60, 60, 60, 60, 60, 65, 65, 65, 65, 70, 70, 70, 75, 75, 75, 80$

Sum of the values: $25 + 40 + (7 \times 50) + (8 \times 60) + (4 \times 65) + (3 \times 70) + (3 \times 75) + 80 = 1595$

There are 27 values.

$\text{Mean} = \frac{1595}{27} \approx 59.04$

Step 2: Subtract the Mean from Each Data Point and Square the Result

Subtract the mean from each data point, then square the result:

$(25 - 59.04)^2, (40 - 59.04)^2, (50 - 59.04)^2, \dots$

Step 3: Find the Variance

Variance is the average of the squared differences from the mean.

$\text{Variance} = \frac{\sum \left( (x_i - \mu)^2 \right)}{N}$

Step 4: Take the Square Root of the Variance

The standard deviation is the square root of the variance.

Now, let me compute the standard deviation using the data you provided:

Calculating all steps:

• Squared deviations
• Sum of squared deviations
• Variance
• Standard deviation

Give me a moment! numpy as np

Given data set

data = [25, 40, 50, 50, 50, 50, 50, 50, 60, 60, 60, 60, 60, 60, 60, 60, 65, 65, 65, 65, 70, 70, 70, 75, 75, 75, 80]

Calculate mean, variance, and standard deviation

mean = np.mean(data) variance = np.var(data) std_deviation = np.sqrt(variance)

mean, variance, std_deviationHere are the calculations for the standard deviation of the data set:

• Mean: 60.0
• Variance: 137.04
• Standard Deviation: 11.71

This means that the data points are spread out with a standard deviation of approximately 11.71 from the mean.

Would you like more details on any of these steps? Or do you have any other questions?

Here are some related questions for further learning:

1. What is the difference between population and sample standard deviation?
2. How can outliers affect the standard deviation?
3. How do you calculate the variance of a data set?
4. What are the applications of standard deviation in real-life scenarios?
5. How is standard deviation used in financial analysis?

Tip: When comparing two data sets, the one with the higher standard deviation generally has more variability or spread.