Problem 10 - Algebra 2

Question

Find the variance and standard deviation of each set of data to the nearest tenth. {234, 345, 123, 368, 279, 876, 456, 235, 333, 444}

Step-by-Step Solution

Answer

Variance: 37685.1, Standard Deviation: 194.1

Step 1: Calculate the Mean

The first step to finding the variance and standard deviation is to calculate the mean (average) of the data set. Add all the numbers together, then divide by the total number of values. \[\text{Mean} = \frac{234 + 345 + 123 + 368 + 279 + 876 + 456 + 235 + 333 + 444}{10}\]Calculating this gives:\[\text{Mean} = \frac{3693}{10} = 369.3\]

Step 2: Calculate Each Deviation from the Mean

Subtract the mean from each data point to find the deviation of each.\[234 - 369.3 = -135.3, \quad 345 - 369.3 = -24.3, \quad 123 - 369.3 = -246.3\]\[368 - 369.3 = -1.3, \quad 279 - 369.3 = -90.3, \quad 876 - 369.3 = 506.7\]\[456 - 369.3 = 86.7, \quad 235 - 369.3 = -134.3, \quad 333 - 369.3 = -36.3\]\[444 - 369.3 = 74.7\]

Step 3: Square Each Deviation

Square each of the deviations you calculated in Step 2 to ensure they are positive.\[(-135.3)^2 = 18305.29, \quad (-24.3)^2 = 590.49, \quad (-246.3)^2 = 60613.69\]\[(-1.3)^2 = 1.69, \quad (-90.3)^2 = 8154.09, \quad (506.7)^2 = 256734.49\]\[(86.7)^2 = 7515.69, \quad (-134.3)^2 = 18037.49, \quad (-36.3)^2 = 1317.69\]\[(74.7)^2 = 5580.09\]

Step 4: Calculate the Variance

To find the variance, average these squared deviations. Add all the squared values calculated in Step 3 and divide by the number of data points (10).\[\text{Variance} = \frac{18305.29 + 590.49 + 60613.69 + 1.69 + 8154.09 + 256734.49 + 7515.69 + 18037.49 + 1317.69 + 5580.09}{10}\]Calculating this gives:\[\text{Variance} = \frac{376850.69}{10} = 37685.1\]

Step 5: Calculate the Standard Deviation

The standard deviation is the square root of the variance. Take the square root of the variance calculated in Step 4.\[\text{Standard Deviation} = \sqrt{37685.1} \approx 194.1\]

Key Concepts

Standard Deviation

When dealing with numerical data, it's helpful to understand how much the values in a data set differ from the average or mean value. This is where the concept of standard deviation comes in. Standard deviation provides a summary of the spread or dispersion of a set of data points relative to its mean. It is represented as the square root of the variance. So, if the variance is large, the data points are spread out over a wide range of values.

• A small standard deviation indicates that the data points tend to be close to the mean.
• A large standard deviation suggests that the data points tend to be spread out over a large range of values.

In the example provided here, the calculated standard deviation is 194.1, meaning the data points are quite spread out from the mean of 369.3.

Data Analysis

Data analysis is a crucial process of inspecting, cleansing, transforming, and modeling data to discover useful information, draw conclusions, and support decision-making. In the context of the exercise, data analysis involves calculating statistical measures like the mean, variance, and standard deviation. These calculations help in understanding the characteristics of the data set.

• Mean: Provides the average value around which all other data points revolve.
• Variance and Standard Deviation: Indicate how spread out the values are around the mean.

By performing these calculations, data analysis enables us to look beyond the surface of the data and find patterns and relationships that can inform decisions or predict future trends.

Mean Deviation

Mean deviation, also known as average deviation, involves finding the average of the absolute differences between each data point and the mean. This measurement provides insight into the dispersion of the data, similar to the variance and standard deviation, but without squaring the differences.

• To calculate mean deviation: Subtract the mean from each data point, take the absolute value of each difference, and then calculate the average of those absolute differences.
• Unlike variance and standard deviation, mean deviation provides a direct measure of deviation in the same units as the data.

Mean deviation, while sometimes less commonly used, offers a straightforward view of how much each data point differs from the central tendency of the data set.

Statistical Variance

Statistical variance is a measure that represents the degree of spread in a set of data points. It measures how far each number in the data set is from the mean, by averaging the squared differences. Understanding variance is crucial as it sets the foundation for calculating standard deviation.

• Variance gives a clear picture of data distribution. High variance indicates that data points are far from the mean, and low variance indicates they are close.
• The unit of variance is always the square of the unit of data, which is why we use standard deviation to bring it back to the same unit.

In the example data set, the variance calculated is 37685.1. This value suggests a substantial spread in the data values around their mean.