Standard Deviation Formula, Statistics, Variance, Sample and Population Mean
Summary
TLDRThis 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
📊 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.
🔢 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.
📉 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.
Mindmap
Keywords
💡Standard Deviation
💡Population Standard Deviation
💡Sample Standard Deviation
💡Population Mean
💡Sample Mean
💡Variance
💡Sigma (Σ)
💡Spread
💡Number Line
💡Data Points
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
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