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## Statistics and probability

### Course: Statistics and probability > Unit 3

Lesson 4: Variance and standard deviation of a population- Measures of spread: range, variance & standard deviation
- Variance of a population
- Population standard deviation
- The idea of spread and standard deviation
- Calculating standard deviation step by step
- Standard deviation of a population
- Mean and standard deviation versus median and IQR
- Concept check: Standard deviation
- Statistics: Alternate variance formulas

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# Concept check: Standard deviation

Six questions that will help you understand standard deviation more deeply.

## Introduction

The questions below are designed to help you think deeply about standard deviation and its formula.

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

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

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

## Part 2

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

## Part 3

**Why or why not?**

*Hint: Think about the formula.*

## 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:

**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:

Here's the formula that statisticians actually use:

**Are the two formulas equivalent?**

## Want to join the conversation?

- I agree with snowball1984...what makes this the preferred measure? It seems that there could also be another way to calculate deviation based on any way to calculate mean. Could we say that there is:

1) a linear mean deviation, [or mean absolute deviation,]

2) a standard deviation, [using the squared differences.]

3) a geometric deviation, [using the cubed differences,]

4) a harmonic deviation, [using a harmonic mean...(21 votes) - It was said that a sample variance is calculated by dividing by n-1. Why is the sample standard deviation here not dividing by n-1?(11 votes)
- It should be.

That being said, in this case, it doesn't matter so much, as Sal is not asking about how the SD will behave when changing a number. So, we're really just interested in whether the numerator is increasing or decreasing, dividing by`n`

or by`n-1`

is only scaling that numerator. So, yes, the formula should have an`n-1`

in the denominator, but having just`n`

there will not change the answer(6 votes)

- It appears this and the previous module are not using the correct variance equation. They should use the population variance equation and not the sample variance equation. N and mu, not n and x-bar.(10 votes)
- Does it make sense to calculate the standard deviation of a collection of complex numbers? That final question in Part 6 makes me wonder.(4 votes)
- Part 3

Can standard deviation be negative?

Hint: Think about the formula.

I thought about the formula and came up with the following.

Since symbol √ always means non-negative square root then SD cannot be negative.

The content of the Explain section then is irrelevant. In fact, it explains why*variance*cannot be negative which is a different question after all.

Correct me if I am wrong.(3 votes) - when using the formula of standard deviation that statitians use how do you know what the sign (negative or positive the answer to the devation is going to be?(2 votes)
- This question is thoroughly answered if you click [Explain] below the last question in the article.(3 votes)

- So the formula for MAD is (∑|x-mean|)/n ? Why not x_i ? I mean, the ith value in the data set.(2 votes)
- There are two ways how we can define this sum. First, we can use indexing as you suggest. Second, we can use ∈ operator to show this is the sum of all elements x such that x ∈ S where S is the set of given data. Here the second way is used with hiding x ∈ S under the sigma symbol.(2 votes)

- Can standard deviation help to check leniency and strictness of three raters?(2 votes)
- can standard deviation help to interpret leniency and strictness of three raters?(2 votes)
- I don't think the explanation in the section 'No, standard deviation cannot be negative!' is useful. It explains very well why what is under the square root sign cannot be negative, but that is not the same thing as the StdDev itself not being negative (just means it is not imaginary I think). I am of the impression that the reason the StdDev will not be negative is because we select the principal (non-negative) square root. Am I missing something, or am I correct in thinking that this section is not quite showing the right thing?(2 votes)
- It's not about the square root. It's because of the squaring of the distances from the mean - the "2" at the top right of the formula. Every square is positive (note that negative times negative is always positive).(1 vote)