Consider The Data Set: 55, 59, 52, 65, 56, 59, 58 What Is The Standard Deviation Of The Data Set? Round Your Answer To The Nearest Tenth. Enter Your Answer In The Box.
Understanding Standard Deviation
Standard deviation is a statistical measure that calculates the amount of variation or dispersion of a set of values. It represents how much the individual data points deviate from the mean value of the dataset. In other words, it measures the spread or dispersion of the data points from the average value. Standard deviation is an essential concept in mathematics, particularly in statistics and data analysis.
Calculating Standard Deviation: A Formula
The formula for calculating standard deviation is:
σ = √[(Σ(xi - μ)²) / (n - 1)]
where:
- σ is the standard deviation
- xi is each individual data point
- μ is the mean of the dataset
- n is the number of data points
- Σ denotes the sum of the values
Step 1: Calculate the Mean
To calculate the standard deviation, we first need to find the mean of the dataset. The mean is calculated by summing up all the data points and dividing by the number of data points.
Dataset: 55, 59, 52, 65, 56, 59, 58
Step 1: Sum up the data points
55 + 59 = 114 114 + 52 = 166 166 + 65 = 231 231 + 56 = 287 287 + 59 = 346 346 + 58 = 404
Step 2: Calculate the mean
404 / 7 = 57.71 (rounded to two decimal places)
Step 2: Calculate the Deviation from the Mean
Now that we have the mean, we can calculate the deviation of each data point from the mean.
| Data Point | Deviation from Mean |
|---|---|
| 55 | 55 - 57.71 = -2.71 |
| 59 | 59 - 57.71 = 1.29 |
| 52 | 52 - 57.71 = -5.71 |
| 65 | 65 - 57.71 = 7.29 |
| 56 | 56 - 57.71 = -1.71 |
| 59 | 59 - 57.71 = 1.29 |
| 58 | 58 - 57.71 = 0.29 |
Step 3: Calculate the Squared Deviation
Next, we need to calculate the squared deviation of each data point from the mean.
| Data Point | Squared Deviation |
|---|---|
| 55 | (-2.71)² = 7.33 |
| 59 | (1.29)² = 1.66 |
| 52 | (-5.71)² = 32.73 |
| 65 | (7.29)² = 53.13 |
| 56 | (-1.71)² = 2.92 |
| 59 | (1.29)² = 1.66 |
| 58 | (0.29)² = 0.08 |
Step 4: Calculate the Sum of Squared Deviation
Now, we need to calculate the sum of the squared deviations.
7.33 + 1.66 = 8.99 8.99 + 32.73 = 41.72 41.72 + 53.13 = 94.85 94.85 + 2.92 = 97.77 97.77 + 1.66 = 99.43 99.43 + 0.08 = 99.51
Step 5: Calculate the Standard Deviation
Finally, we can calculate the standard deviation using the formula:
σ = √[(Σ(xi - μ)²) / (n - 1)]
σ = √(99.51 / (7 - 1)) σ = √(99.51 / 6) σ = √16.585 σ = 4.08 (rounded to the nearest tenth)
Conclusion
Q: What is standard deviation?
A: Standard deviation is a statistical measure that calculates the amount of variation or dispersion of a set of values. It represents how much the individual data points deviate from the mean value of the dataset.
Q: Why is standard deviation important?
A: Standard deviation is an essential concept in mathematics, particularly in statistics and data analysis. It helps to understand the spread or dispersion of the data points from the average value, which is crucial in making informed decisions in various fields such as finance, economics, and social sciences.
Q: How is standard deviation calculated?
A: The formula for calculating standard deviation is:
σ = √[(Σ(xi - μ)²) / (n - 1)]
where:
- σ is the standard deviation
- xi is each individual data point
- μ is the mean of the dataset
- n is the number of data points
- Σ denotes the sum of the values
Q: What is the difference between population standard deviation and sample standard deviation?
A: Population standard deviation is calculated when the entire population is known, whereas sample standard deviation is calculated when a sample of the population is used. The formula for sample standard deviation is:
σ = √[(Σ(xi - μ)²) / (n - 1)]
where:
- σ is the sample standard deviation
- xi is each individual data point in the sample
- μ is the mean of the sample
- n is the number of data points in the sample
- Σ denotes the sum of the values
Q: What is the significance of the standard deviation value?
A: A standard deviation value close to zero indicates that the data points are closely clustered around the mean, whereas a standard deviation value far from zero indicates that the data points are spread out over a larger range.
Q: How is standard deviation used in real-life scenarios?
A: Standard deviation is used in various fields such as finance, economics, and social sciences to:
- Analyze stock market performance
- Understand economic trends
- Evaluate the effectiveness of a treatment in medicine
- Determine the reliability of a survey
Q: Can standard deviation be negative?
A: No, standard deviation cannot be negative. The standard deviation value is always positive, as it represents the amount of variation or dispersion of the data points from the mean value.
Q: Can standard deviation be zero?
A: Yes, standard deviation can be zero. This occurs when all the data points are identical, resulting in no variation or dispersion from the mean value.
Q: How is standard deviation related to variance?
A: Standard deviation is the square root of variance. Variance is the average of the squared differences from the mean, whereas standard deviation is the square root of this value.
Q: Can standard deviation be used to compare different datasets?
A: Yes, standard deviation can be used to compare different datasets. By comparing the standard deviation values of two or more datasets, you can determine which dataset has more variation or dispersion from the mean value.
Q: What are some common applications of standard deviation?
A: Some common applications of standard deviation include:
- Analyzing stock market performance
- Understanding economic trends
- Evaluating the effectiveness of a treatment in medicine
- Determining the reliability of a survey
- Comparing the performance of different products or services
Conclusion
In this article, we have answered some frequently asked questions about standard deviation, including its definition, importance, calculation, and applications. We have also discussed the significance of standard deviation values and how they can be used to compare different datasets.