standard deviation

NOVEMBER 14, 2023

What is standard deviation in math? Definition.

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. It provides a way to understand how spread out the data is from the mean or average value. In other words, it measures the average distance between each data point and the mean.

History of standard deviation.

The concept of standard deviation was introduced by the English mathematician Karl Pearson in the late 19th century. Pearson developed the concept as a way to measure the variability in data sets and to better understand the distribution of data. Since then, standard deviation has become a fundamental tool in statistics and is widely used in various fields such as finance, economics, and social sciences.

What grade level is standard deviation for?

Standard deviation is typically introduced in high school mathematics courses, particularly in statistics or advanced algebra classes. It is also covered in college-level courses in statistics, probability, and data analysis.

What knowledge points does standard deviation contain? And detailed explanation step by step.

To understand standard deviation, it is important to have a basic understanding of the following concepts:

  1. Mean: The average value of a set of data points.
  2. Variance: The average of the squared differences between each data point and the mean.
  3. Square root: The mathematical operation that gives the non-negative value whose square is equal to a given number.

Step-by-step explanation of calculating standard deviation:

  1. Calculate the mean of the data set.
  2. Subtract the mean from each data point and square the result.
  3. Calculate the average of the squared differences obtained in step 2.
  4. Take the square root of the result obtained in step 3.

Types of standard deviation.

There are two main types of standard deviation:

  1. Population Standard Deviation: This is used when the entire population is known and all its data points are included in the calculation. It is denoted by the symbol σ (sigma).

  2. Sample Standard Deviation: This is used when only a subset or sample of the population is known. It is denoted by the symbol s.

Properties of standard deviation.

Some important properties of standard deviation include:

  1. Standard deviation is always non-negative.
  2. If all data points are the same, the standard deviation is zero.
  3. Standard deviation is sensitive to outliers or extreme values in the data set.
  4. Adding or subtracting a constant value to each data point does not change the standard deviation.

How to find or calculate standard deviation?

To calculate the standard deviation, follow these steps:

  1. Calculate the mean of the data set.
  2. Subtract the mean from each data point and square the result.
  3. Calculate the average of the squared differences obtained in step 2.
  4. Take the square root of the result obtained in step 3.

What is the formula or equation for standard deviation?

The formula for calculating standard deviation is as follows:

For population standard deviation (σ): σ = √(Σ(x - μ)² / N)

For sample standard deviation (s): s = √(Σ(x - x̄)² / (n - 1))

Where:

  • x represents each data point in the set.
  • μ (mu) represents the population mean.
  • x̄ (x-bar) represents the sample mean.
  • N represents the total number of data points in the population.
  • n represents the total number of data points in the sample.

How to apply the standard deviation formula or equation?

To apply the standard deviation formula, substitute the values of each data point, mean, and the number of data points into the respective formula for population or sample standard deviation. Then, perform the necessary calculations to obtain the final result.

What is the symbol or abbreviation for standard deviation?

The symbol or abbreviation for population standard deviation is σ (sigma), while the symbol or abbreviation for sample standard deviation is s.

What are the methods for standard deviation?

There are several methods for calculating standard deviation, including:

  1. Direct Calculation: This involves manually calculating the mean, squared differences, and taking the square root.

  2. Using Statistical Software: Many statistical software packages, such as Excel, SPSS, or R, have built-in functions to calculate standard deviation.

  3. Using Calculators: Some scientific or graphing calculators also have functions to calculate standard deviation.

More than 3 solved examples on standard deviation.

Example 1: Consider the following data set: 5, 8, 10, 12, 15. Calculate the population standard deviation.

Solution: Step 1: Calculate the mean: (5 + 8 + 10 + 12 + 15) / 5 = 10. Step 2: Subtract the mean from each data point and square the result: (5 - 10)² = 25, (8 - 10)² = 4, (10 - 10)² = 0, (12 - 10)² = 4, (15 - 10)² = 25. Step 3: Calculate the average of the squared differences: (25 + 4 + 0 + 4 + 25) / 5 = 10.8. Step 4: Take the square root of the result: √10.8 ≈ 3.29. Therefore, the population standard deviation is approximately 3.29.

Example 2: Consider the following data set: 12, 15, 18, 21, 24. Calculate the sample standard deviation.

Solution: Step 1: Calculate the mean: (12 + 15 + 18 + 21 + 24) / 5 = 18. Step 2: Subtract the mean from each data point and square the result: (12 - 18)² = 36, (15 - 18)² = 9, (18 - 18)² = 0, (21 - 18)² = 9, (24 - 18)² = 36. Step 3: Calculate the average of the squared differences: (36 + 9 + 0 + 9 + 36) / 4 = 22.5. Step 4: Take the square root of the result: √22.5 ≈ 4.74. Therefore, the sample standard deviation is approximately 4.74.

Example 3: Consider the following data set: 2, 4, 6, 8, 10. Calculate the population standard deviation.

Solution: Step 1: Calculate the mean: (2 + 4 + 6 + 8 + 10) / 5 = 6. Step 2: Subtract the mean from each data point and square the result: (2 - 6)² = 16, (4 - 6)² = 4, (6 - 6)² = 0, (8 - 6)² = 4, (10 - 6)² = 16. Step 3: Calculate the average of the squared differences: (16 + 4 + 0 + 4 + 16) / 5 = 8. Step 4: Take the square root of the result: √8 ≈ 2.83. Therefore, the population standard deviation is approximately 2.83.

Practice Problems on standard deviation.

  1. Calculate the population standard deviation for the following data set: 3, 6, 9, 12, 15.

  2. Calculate the sample standard deviation for the following data set: 20, 25, 30, 35, 40.

  3. Calculate the population standard deviation for the following data set: 1, 3, 5, 7, 9.

FAQ on standard deviation.

Question: What is standard deviation? Answer: Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values.

Question: How is standard deviation calculated? Answer: Standard deviation is calculated by finding the mean of the data set, subtracting the mean from each data point, squaring the result, calculating the average of the squared differences, and taking the square root of the result.

Question: What is the difference between population and sample standard deviation? Answer: Population standard deviation is used when the entire population is known, while sample standard deviation is used when only a subset or sample of the population is known.

Question: What are the properties of standard deviation? Answer: Some properties of standard deviation include being non-negative, sensitive to outliers, and unaffected by adding or subtracting a constant value to each data point.

Question: What grade level is standard deviation for? Answer: Standard deviation is typically introduced in high school mathematics courses, particularly in statistics or advanced algebra classes. It is also covered in college-level courses in statistics, probability, and data analysis.