Standard Deviation Formula, Statistics, Variance, Sample and Population Mean

Summary

This educational video script explains the process of calculating standard deviation, both for a population and a sample. It introduces two formulas: one for population standard deviation (using the population mean, μ, and dividing by n) and another for sample standard deviation (using the sample mean and dividing by n-1). The script uses the example of two sets of numbers, 4, 5, 6 and 3, 5, 7, to demonstrate the calculation, highlighting that the set with more spread out numbers will have a higher standard deviation. It also touches on the concept of variance, which is the square of the standard deviation. The video concludes by inviting viewers to explore more educational content on the creator's channel and website.

Takeaways

• 📐 The video explains how to calculate standard deviation, which measures how spread out numbers are in a set.
• 🔢 There are two formulas for standard deviation: one for the entire population and one for a sample of the population.
• 👫 The population standard deviation (σ) is calculated by summing the squared differences from the mean (μ), divided by the number of data points (n), and then taking the square root.
• 🔄 For a sample standard deviation (s), the formula is similar but divides by n-1 instead of n, accounting for the sample size.
• 📊 The video uses an example with the sets 4, 5, 6 and 3, 5, 7 to illustrate that the latter has a greater standard deviation due to its numbers being more spread out.
• 🧮 To calculate the mean, sum all numbers and divide by the count, which for evenly spaced numbers will be the middle value.
• 📉 The process for calculating standard deviation involves subtracting the mean from each number, squaring the result, summing these squares, dividing by n (or n-1), and taking the square root.
• 🔍 The video demonstrates the calculation for the set 3, 5, 7, resulting in a standard deviation of approximately 1.63.
• 📈 The variance is the square of the standard deviation and is calculated by summing the squared differences from the mean and dividing by n (or n-1), without taking the square root.
• 🌐 The video concludes by mentioning that the process for calculating population and sample standard deviation is similar, with the main difference being the divisor (n vs. n-1).

Q & A

What is the population standard deviation formula?

The population standard deviation formula is represented by sigma (σ). It is equal to the square root of the sum of the squared differences between each data point and the population mean (μ), divided by the total number of data points (N).

How does the sample standard deviation formula differ from the population standard deviation formula?

The sample standard deviation formula differs from the population standard deviation in that it divides by (N - 1) instead of N. This adjustment is made to account for the fact that the data set is only a sample of the entire population.

What does standard deviation measure?

Standard deviation measures how spread out the numbers in a data set are in relation to each other. A higher standard deviation indicates that the numbers are more spread out, while a lower standard deviation indicates that the numbers are closer together.

Why does the set of numbers 3, 5, and 7 have a higher standard deviation than the set 4, 5, and 6?

The set of numbers 3, 5, and 7 has a higher standard deviation because the numbers are more spread out compared to the set 4, 5, and 6. The differences between each number and the mean are larger in the first set.

How do you calculate the mean of a data set?

To calculate the mean of a data set, sum all the numbers in the set and then divide by the total number of data points.

What are the steps to calculate the population standard deviation?

First, calculate the mean of the data set. Then, subtract the mean from each data point and square the result. Sum all the squared differences, divide by the total number of data points (N), and finally, take the square root of the result.

How would you calculate the standard deviation for the numbers 3, 5, and 7?

First, calculate the mean, which is 5. Then, find the differences between each number and the mean, square those differences, sum them, and divide by the number of data points (3). Finally, take the square root of the result, which gives a standard deviation of approximately 1.63.

What is the variance and how is it related to standard deviation?

Variance is the square of the standard deviation. It measures the average of the squared differences from the mean. For example, if the standard deviation is 1.63, the variance would be 1.63 squared, approximately 2.66.

Why do you use N - 1 in the sample standard deviation formula instead of N?

N - 1 is used in the sample standard deviation formula to correct for bias in the estimation of the population variance. This adjustment, known as Bessel's correction, accounts for the fact that a sample may not fully represent the entire population.

What subjects does the instructor offer on their website?

The instructor offers tutorials on algebra, trigonometry, pre-calculus, calculus, chemistry, and physics on their website.

Outlines

00:00

📊 Introduction to Standard Deviation Calculation

This paragraph introduces the concept of standard deviation, a measure used to quantify the amount of variation or dispersion in a set of values. It distinguishes between two formulas: the population standard deviation (σ) and the sample standard deviation (s). The population standard deviation is calculated as the square root of the sum of the squared differences between each data point and the population mean (μ), divided by the number of data points (n). The sample standard deviation formula is similar but divides by n-1 instead of n. The paragraph sets the stage for an example to compare the standard deviation of two sets of numbers: {4, 5, 6} and {3, 5, 7}, aiming to demonstrate which set has a greater standard deviation.

05:00

🔢 Calculating Standard Deviation: An Example Walkthrough

This paragraph delves into a step-by-step calculation of the standard deviation for the set {3, 5, 7} using the population standard deviation formula. It begins by calculating the mean of the set, which is the middle number, 5. Next, it guides through the process of finding the differences between each data point and the mean, squaring these differences, summing them, and then dividing by the number of data points (n=3). The result is then used to calculate the standard deviation by taking the square root of the sum divided by n. The calculated standard deviation for this set is approximately 1.63, illustrating a methodical approach to understanding the concept.

10:02

📉 Comparing Standard Deviations and Calculating Variance

The final paragraph compares the standard deviation of the set {4, 5, 6} with the previously calculated set {3, 5, 7}. It reinforces the concept that standard deviation measures how spread out numbers are, with the set {4, 5, 6} having a lower standard deviation due to its numbers being closer to each other. The mean for this set is also calculated as 5, and the standard deviation is computed to be approximately 0.816, which is lower than that of the set {3, 5, 7}. The paragraph concludes with an explanation of how to calculate variance, which is the square of the standard deviation, for the set {3, 5, 7}. It also directs viewers to the tutor's website for more educational content on various subjects.

Keywords

💡Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the video, it is used to determine how spread out a set of numbers is. The video explains two formulas for calculating standard deviation: one for the entire population and one for a sample of the population. The script uses the example of the numbers 3, 5, and 7 to demonstrate the calculation of standard deviation, which is approximately 1.63, indicating the spread of these numbers from their mean.

💡Population Standard Deviation

Population standard deviation refers to the calculation of standard deviation when considering the entire population. It is represented by the Greek letter sigma (σ) and is calculated by dividing the sum of the squared differences between each data point and the population mean by the number of data points (n). The video script provides a formula and an example to illustrate how to calculate it, emphasizing its use when you have data from the whole population.

💡Sample Standard Deviation

Sample standard deviation is used when calculating the standard deviation from a subset of a larger population. It is denoted by 's' and is calculated similarly to the population standard deviation but with a slight difference: the sum of squared differences is divided by n-1 (the number of data points minus one) instead of n. This adjustment accounts for the fact that the sample may not be entirely representative of the population.

💡Population Mean

The population mean, represented by the Greek letter mu (μ), is the average of all the values in a population. In the video, the script explains that to find the population mean, you sum all the numbers in the dataset and divide by the total number of values. It is used as a reference point in the calculation of standard deviation.

💡Sample Mean

Sample mean is the average of the values in a sample taken from a larger population. It is calculated in the same way as the population mean but is used when you only have a subset of the data. The video script uses the sample mean in the calculation of sample standard deviation, emphasizing its role in estimating the population mean from sample data.

💡Variance

Variance is a measure of dispersion that indicates how much the values in a dataset vary from the mean. It is calculated as the average of the squared differences from the mean. In the video, variance is described as the square of the standard deviation, which provides a measure of the spread of the data points without the units being 'standard deviations' from the mean.

💡Sigma (Σ)

In the context of the video, sigma (Σ) is used as a symbol to represent the sum of a series of values, particularly in the calculation of standard deviation and variance. It is a mathematical symbol that instructs to add up all the terms in a sequence, which is a crucial step in both population and sample standard deviation calculations.

💡Spread

Spread refers to the extent to which the values in a dataset are dispersed or spread out from the mean. The video uses the concept of spread to explain the intuition behind standard deviation. A greater spread indicates a higher standard deviation, which means the data points are farther from the mean, as illustrated with the sets 3, 5, 7 and 4, 5, 6.

💡Number Line

A number line is a visual representation of numbers in a straight line, where each point corresponds to a number. In the video, the number line is used to graphically represent the sets of numbers to compare their spread. The script describes how the numbers 4, 5, and 6 are closer together on a number line compared to 3, 5, and 7, helping to visually demonstrate the concept of standard deviation.

💡Data Points

Data points are individual values in a dataset. The video script refers to data points when explaining how to calculate the mean and standard deviation. Each data point is considered in relation to the mean to determine the differences that contribute to the calculation of standard deviation.

Highlights

Introduction to calculating standard deviation with two formulas: population and sample standard deviation.

Explanation of the population standard deviation formula represented by sigma.

Description of the sample standard deviation formula, using 's' instead of sigma.

Clarification that sample standard deviation is calculated with n-1 instead of n.

Discussion on the concept of standard deviation as a measure of how spread out numbers are.

Visual comparison of two sets of numbers to intuitively determine which has a higher standard deviation.

Calculation of the mean for the set of numbers 3, 5, and 7.

Step-by-step calculation of the population standard deviation for the set 3, 5, and 7.

Explanation of the process to calculate standard deviation using the formula.

Calculation of the variance as the square of the standard deviation.

Instruction to pause the video for the audience to calculate standard deviation for the set 4, 5, and 6.

Calculation of the mean for the set of numbers 4, 5, and 6.

Step-by-step calculation of the population standard deviation for the set 4, 5, and 6.

Comparison of standard deviations between the two sets of numbers to illustrate the concept.

Conclusion of the video with a summary of the methods to calculate population and sample standard deviation and variance.

Promotion of the video creator's channel and website for more educational content.

Transcripts

in this video we're going to calculate

the standard deviation of a set of

numbers

now there's two formulas you need to be

aware of the first one

is the population

standard deviation

now this formula

is represented by the letter sigma

that's the standard deviation

it's equal to the sum

of all the differences between

every point

in the data set and the population mean

the population mean is mu

which is this symbol here

and then you need to square it

divided by

n which is

all of the

numbers in the set

and then you got to take the square root

of the whole result

so that's the population standard

deviation

the next formula is the sample standard

deviation

so let's say if

you have

just a sample of a population not the

entire population

if you just have a sample data out of

the entire data

then you want to use this formula

s which is the standard deviation

is equal to sigma

the sum of all of the

differences between every point

and the mean that's the sample mean

in the other equation we had the

population mean represented by mu

but this is the sample mean which is

basically the average of all the data

points in the set

and then you have to square it but it's

going to be divided by n

minus 1 as opposed to n

and so that's how you calculate the

standard deviation of the sample

now let's work on an example

let's say if we have two set of numbers

four five and six

and also three

five and seven

which one has a greater standard

deviation let's use the population

standard deviation formula

but if we had to guess

which set of numbers

has the greater standard deviation is it

the one on the left or the one on the

right

what would you say

we need to understand the basic idea

of standard deviation you need to know

what it measures

standard deviation tells you

how far apart the numbers

are related to each other

so the more spread out they are the

greater the standard deviation

four five and six are closer to each

other than three five and seven

and you could tell if you plot them on a

number line

let's put five in the middle

so 4 5 and 6

here they are on a number line

now in contrast

let's put the same numbers

on this number line

we're going to have 3

5 and 7.

so if you look at the the red points

the points in red are further apart

the points in blue they're very close

together

so therefore

four five and six

has a lower standard deviation in three

five and seven

so sigma is low

and here the sigma value is high

now go ahead and calculate the

standard deviation

for this

for the set of numbers three five and

seven

so what's the first thing that we should

do

the first thing that we should do is

calculate the mean

to find the mean

it's going to be the sum

of all the numbers divided by 3.

now because the three numbers are evenly

spaced apart the mean is going gonna be

the middle number five

three plus five is eight eight plus

seven

is fifteen

fifteen divided by three is five so

that's the mean

now what should we do next now that we

have the mean

now think of the formula

it's going to be a sigma

of every point minus the

mean squared

divided by

n

and then all of this is within the

square root

so here's how to use the equation first

we're going to use the first point 3

subtract it by the mean and then squared

next we're going to take the second

point 5

subtract it from the mean squared

and then it's going to be 7

minus 5

squared

so each of these three points

you're going to plug into x sub i

and then you're going to square the

differences between each of those values

and the sigma represents sum so you're

going to add every difference

that you get

or you can add the square of every

difference that you get

and now let's divide it by n

so n is the number of numbers that we

have in this set there are three numbers

inside

so n is 3

and then we're going to take the square

root of the entire thing

three minus five is negative two

negative two squared is four

five minus five is zero

seven minus five is two two squared is

four

four plus four is eight

so we have the square root of eight

divided by three

and at this point we're going to use the

calculator

8 divided by 3 is about 2.67 and if you

take the square root of that

you're going to get 1.63

so that's the standard deviation

for 3 5 and 7.

now let's calculate the standard

deviation

for the other set of numbers four five

and six so why don't you go ahead and

pause the video

and try this example

calculate the standard deviation using

the same formula

so let's go ahead and begin let's

calculate the population mean

it's going to be 4 plus 5 plus 6 divided

by

the number of numbers that we have which

is

3. four plus six is ten ten plus five is

fifteen and we know that fifteen divided

by three is five

so once again any time the numbers are

evenly spread apart the mean is going to

be the middle number

so now

we can calculate the standard deviation

so sigma

is going to equal the square root but

before we do that let's calculate

the differences

so the first difference that we have the

first number is going to be 4

and we're going to subtract it from the

mean

and then square it the next number

is 5

subtract it from the mean

and then square it

and then after that the last number is

six

this is going to be

six minus five squared

now it's divided by n

and let's not forget to take the square

root of the entire thing

four minus five is negative one negative

one squared is simply one

five minus five is zero

six minus five is one

and it's all divided by three one plus

one is two

so we have the square root of two

divided by 3.

now 2 divided by 3 as a decimal is about

0.67

and the square root of 0.67

is 0.816

so as you can see the standard deviation

is less because

these numbers are closer to each other

they're not far apart from the mean

in the other example

three five and seven

they're further apart from the mean

which is five

three is two units away from five

four

is only one unit away from five and

that's why the standard deviation is so

much less

now let's go back to the first example

we said that the population standard

deviation

is approximately 1.63

so given this information

how can you calculate

the variance

v a r i a-n-c-e

how can we find the variance

the variance is simply the square

of the standard deviation

so 1.63 squared

is equal to now keep in mind this is a

rounded answer

i don't remember what the exact answer

was but

once you square it it's about 2.66

so that's how you can calculate the

variance

the formula for variance is basically

the sum

of all the square differences

between every point and the population

mean

divided by n

it's basically the same formula without

the square root symbol

well that's it for this video so now you

know how to calculate

the population standard deviation and

also the sample standard deviation even

though we did just one of them the

process is the same

of finding the other one

the only difference is you have n minus

one instead of n

you also know how to calculate the

variance as well

so that concludes this video by the way

if you want to find more of my videos

you can check out my channel or visit my

website video.tutor.net and you can find

playlists on algebra trade pre-calculus

calculus

chemistry and physics

so those are the subjects that i

currently offer right now

and if you're interested just feel free

to check that out

so thanks again for watching