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Standard Deviation

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Standard Deviation

Study Guide

Key Definition

Standard deviation is the measure of how spread out your data is. It indicates how closely data is gathered around the mean. A smaller standard deviation means data is closer to the mean.

Important Notes

Standard deviation is a form of dispersion.
A high standard deviation means data points are spread out.
In business, a smaller standard deviation is often preferred.
Do not round intermediate results when calculating.
Square all deviations before summing.

Mathematical Notation

$\sum$ represents summation

$x - \bar{x}$ represents deviation from the mean

$x^2$ represents squaring a number

$\sqrt{x}$ represents the square root

$\frac{a}{b}$ represents division

Remember to use proper notation when solving problems

Why It Works

Standard deviation provides a measure of the average distance of each data point from the mean, helping to understand the data's variability.

Remember

For sample standard deviation (s), divide by $n - 1$; for population standard deviation (σ), divide by $n$. The formula involves finding the mean, calculating deviations, squaring them, summing, dividing, and taking the square root.

Quick Reference

Variance:$\frac{\sum (x - \bar{x})^2}{n - 1}$

Understanding Standard Deviation

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Video explanation of this concept

Beginner

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Beginner Explanation

Standard deviation quantifies how much data varies around the mean. For a sample of data, we use the sample standard deviation formula: $s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If the mean of a data set is $5$ and the data points are $3, 5, 7$, what is the deviation of $3$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You have test scores: $85, 90, 95$. Calculate the standard deviation.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given data $2, 4, 4, 4, 5, 5, 7, 9$, calculate the standard deviation.

Challenge Quiz

Single Choice Quiz

Advanced

For data set $10, 12, 23, 23, 16, 23, 21, 16$, what is the variance?

Recap

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