Planets The planets of the solar system have the numbers of moons listed below in order from the sun. (Pluto is not included because it was uninvited from the solar system party in 2006.) Include appropriate units whenever relevant.

0 0 1 2 17 28 21 8

e. Find the standard deviation.
f. Find the variance.

Verified step by step guidance

Step 1: Begin by identifying the data set provided, which represents the number of moons for each planet in the solar system: {0, 0, 1, 2, 17, 28, 21, 8}. Note that these values are the raw data points.

Step 2: Calculate the mean (average) of the data set. The formula for the mean is:

μ = \frac{\sum x}{n}

, where

x

represents each data point and

n

is the total number of data points.

Step 3: Compute the squared differences between each data point and the mean. For each data point

x

, calculate:

{x - μ}^{2}

. This step is essential for both variance and standard deviation calculations.

Step 4: Find the variance using the formula:

σ 2 = \frac{\sum {x - μ}^{2}}{n}

. This represents the average of the squared differences.

Step 5: Calculate the standard deviation by taking the square root of the variance. The formula is:

σ = \sqrt{\frac{\sum {x - μ}^{2}}{n}}

. This provides a measure of the spread of the data points around the mean.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It quantifies how much the individual data points differ from the mean of the dataset. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation suggests a wider spread of values. It is commonly used in statistics to understand the distribution of data.

Variance

Variance is the average of the squared differences from the mean, providing a measure of how much the data points in a dataset vary. It is calculated by taking the mean of the squared deviations from the mean. Variance is a foundational concept in statistics, as it helps to understand the spread of data and is directly related to the standard deviation, which is the square root of variance.

Data Set

A data set is a collection of related values or observations that can be analyzed statistically. In this context, the data set consists of the number of moons for each planet in the solar system. Understanding the structure and characteristics of the data set is crucial for calculating statistical measures like variance and standard deviation, as these calculations depend on the specific values within the set.

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