Question

Find the standard deviation of the data set. $$ 20,30,40,80,130 $$

Short answer

Question: Calculate the standard deviation of the following data set: 20, 30, 40, 80, 130. Answer: To calculate the standard deviation, follow these steps: 1. Find the mean of the data set: (20+30+40+80+130)/5 = 60 2. Calculate the differences between each data point and the mean: (-40, -30, -20, 20, 70) 3. Square the differences: (1600, 900, 400, 400, 4900) 4. Find the mean of the squared differences: (1600+900+400+400+4900)/5 = 1840 5. Take the square root of the mean of the squared differences: √1840 ≈ 42.9 The standard deviation of the given data set is approximately 42.9.

Step-by-step solution

Find the mean of the data set.

To find the mean (average) of the data set, sum up all the data points and divide by the total number of data points. Mean=\((20+30+40+80+130)/5\)

Calculate the differences between each data point and mean.

For each data point in the data set, subtract the mean value calculated in Step 1. \((20-Mean),(30-Mean),(40-Mean),(80-Mean),(130-Mean)\)

Square the differences.

Square each difference calculated in Step 2. \(((20-Mean)^2, (30-Mean)^2, (40-Mean)^2, (80-Mean)^2, (130-Mean)^2)\)

Find the mean of squared differences.

Add the squared differences from Step 3, then divide by the total number of data points to get the mean of squared differences. Variance \(= [(20-Mean)^2+(30-Mean)^2+(40-Mean)^2+(80-Mean)^2+(130-Mean)^2]/5\)

Find the square root of the mean of squared differences.

Take the square root of the variance calculated in Step 4 to get the standard deviation of the data set. Standard Deviation \(= \sqrt{Variance}\)

Key concepts

Mean

The mean, often called the average, is a fundamental concept in statistics. It provides a central point that represents a data set effectively.
To find the mean, you simply add up all the numbers in your data set and divide by the total number of data points.
In our example:

• The numbers are 20, 30, 40, 80, and 130.
• First, add them up: 20 + 30 + 40 + 80 + 130 = 300.
• Next, count the data points, which are 5 in this case.
• Finally, divide the sum by the count: \( \frac{300}{5} = 60 \).

Thus, the mean is 60. This number serves as a baseline to measure how spread out the data points are relative to the center.

Variance

Variance is a measure of how much the data points in a data set deviate from the mean. It gives us an idea of the spread or dispersion of the data.
Here's how you calculate it:

• Start by determining the difference between each data point and the mean.
• Square each of these differences to eliminate negative values and emphasize larger differences.
• For the data set 20, 30, 40, 80, 130 with a mean of 60:
• The differences are \((20-60), (30-60), (40-60), (80-60), (130-60)\) or negative 40, negative 30, negative 20, 20, and 70 respectively.
• These squared values are \( (-40)^2, (-30)^2, (-20)^2, 20^2, 70^2\) = 1600, 900, 400, 400, and 4900.
• Add the squared differences: 1600 + 900 + 400 + 400 + 4900 = 8200.
• Then, divide by the count of data points to find the variance: \( \frac{8200}{5} = 1640 \).

Therefore, the variance is 1640, which tells us how much the data points are spread around the mean.

Data Set Analysis

Analyzing a data set involves using several statistical tools to understand its characteristics. A vital part of this process includes calculating measures like the mean, variance, and standard deviation.
Data set analysis can help in identifying trends, making predictions, and drawing conclusions. Here's a simple guide on what to focus on:

• Mean: It provides a central value, helping to understand the average behavior of the data.
• Variance and Standard Deviation: These measure the degree of variation or dispersion around the mean. A small variance indicates that the data points are close to the mean, while a large variance suggests they are spread out.

The example data set, with a mean of 60 and a variance of 1640, shows variability in values such as 20 or 130, which are distanced from the mean. This understanding can aid in making more informed decisions based on the data's behavior.