Variance is a measure of how much the values in a data set differ from the mean (average) value. It is a measure of the spread or dispersion of the data.

The formula to calculate the variance of a data set is:

$\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}$

Where:

- $\sigma^2$ = variance
- $x_i$ = individual data values
- $\mu$ = mean of the data set
- $n$ = number of data values

To calculate the variance, follow these steps:

- Find the mean of the data set.
- Subtract the mean from each data value, then square the result.
- Find the sum of all the squared differences.
- Divide the sum by the number of data values to get the variance.

Properties of Variance:

- Variance is always non-negative.
- If all the data values are identical, the variance is 0.
- If the data values are spread out, the variance is high.

It's important to understand variance because it helps in understanding the data distribution and making decisions based on the variability of the data.

Study Guide:

- Understand the concept of mean and how it is calculated.
- Learn the formula for variance and understand each component of the formula.
- Practice calculating variance for different data sets.
- Understand the properties and significance of variance in real-world scenarios.
- Work on solving problems and applying variance in different contexts.

By understanding and practicing variance, you'll be able to analyze and interpret the spread of data in a meaningful way.

.Study GuideSimilarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

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Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Apply transformations and use symmetry to analyze mathematical situations.

Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.

Measurement (NCTM)

Apply appropriate techniques, tools, and formulas to determine measurements.

Solve problems involving scale factors, using ratio and proportion.

Grade 8 Curriculum Focal Points (NCTM)

Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle

Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.