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## Statistics and probability

### Course: Statistics and probability>Unit 3

Lesson 4: Variance and standard deviation of a population

# Concept check: Standard deviation

## Introduction

Unlike most questions on Khan Academy, some of these questions aren't graded by a computer. You'll learn the most if you try answering each question yourself before clicking "explain".

## The formula (for reference)

The formula for standard deviation (SD) is
start text, S, D, end text, equals, square root of, start fraction, sum, start subscript, end subscript, start superscript, end superscript, open vertical bar, x, minus, x, with, \bar, on top, close vertical bar, squared, divided by, n, end fraction, end square root
where sum means "sum of", x is a value in the data set, x, with, \bar, on top is the mean of the data set, and n is the number of values in the data set.

## Part 1

Consider the simple data set left brace, 1, comma, 4, comma, start color #e84d39, 7, end color #e84d39, comma, 2, comma, 6, right brace.
How does the standard deviation change when start color #e84d39, 7, end color #e84d39 is replaced with start color #1fab54, 12, end color #1fab54?

How can we see this in the formula for standard deviation?

## Part 2

Is it possible to create a data set with 4 data points that has a standard deviation of 0?

If it is possible, do it! Can you create two different data sets? How about three?

## Part 3

Can standard deviation be negative?

Why or why not?

## Part 4

Standard deviation is a measure of spread of a data distribution.
What do you think deviation means?

### Part 5

Here are the formulas for standard deviation (SD) and the formula for mean absolute deviation (MAD), both of which are measures of spread:
start text, S, D, end text, equals, square root of, start fraction, sum, start subscript, end subscript, start superscript, end superscript, open vertical bar, x, minus, x, with, \bar, on top, close vertical bar, squared, divided by, n, end fraction, end square root
start text, M, A, D, end text, equals, start fraction, sum, start subscript, end subscript, start superscript, end superscript, open vertical bar, x, minus, x, with, \bar, on top, close vertical bar, divided by, n, end fraction
What are the similarities between the formulas? What are the differences?

### Part 6

Here's the formula that we've been using to calculate standard deviation:
square root of, start fraction, sum, start subscript, end subscript, start superscript, end superscript, open vertical bar, x, minus, x, with, \bar, on top, close vertical bar, squared, divided by, n, end fraction, end square root
Here's the formula that statisticians actually use:
square root of, start fraction, sum, start subscript, end subscript, start superscript, end superscript, left parenthesis, x, minus, x, with, \bar, on top, right parenthesis, squared, divided by, n, end fraction, end square root
Are the two formulas equivalent?