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

00:10:21
https://www.youtube.com/watch?v=Uk98hiMQgN0

概要

TLDRThis video provides an educational overview on how to calculate standard deviation, a statistical measure of the dispersion of data points in a set. The tutorial describes formulas for both population and sample standard deviations, emphasizing the difference in denominator: 'n' for population and 'n-1' for sample. Using examples like data sets (3, 5, 7) and (4, 5, 6), the video demonstrates calculating mean first, followed by each data point's squared deviation from the mean, and finally computing the square root of the averaged squared deviations. By comparing these data sets, the video illustrates how more scattered numbers yield a higher standard deviation. Additionally, the video explains that variance, another measure of dispersion, is the square of the standard deviation. The presenter also invites viewers to explore more educational content on their channel, covering mathematics to science disciplines.

収穫

  • 📊 Introduction to standard deviation and its formulas for population and sample data.
  • 📐 The difference between population (using 'n') and sample (using 'n-1') standard deviation formulas.
  • 🧮 Step-by-step calculation of standard deviation using example datasets.
  • 📉 Understanding that greater data spread leads to higher standard deviation.
  • 🔍 Variance is derived by squaring the standard deviation.
  • 👥 Comparison of different datasets to illustrate concept application.
  • ↔️ Description of calculating the mean as an initial step in finding standard deviation.
  • 🎓 Explanation of sigma and mu symbols in the context of statistics.
  • 🔢 Problem-solving examples with numbers like (3, 5, 7) vs. (4, 5, 6).
  • 📚 Invitation to explore further educational content in math and science.

タイムライン

  • 00:00:00 - 00:05:00

    The video explains how to calculate the standard deviation for a set of numbers, starting with two key formulas: the population standard deviation and the sample standard deviation. The population standard deviation, represented by the Greek letter sigma, involves finding the differences between each data point and the population mean (mu), squaring these differences, summing them, dividing by the number of data points (n), and then taking the square root. The sample standard deviation is similar but uses the sample mean and divides by n-1 instead of n. An example is provided with two sets of numbers, 4, 5, 6 and 3, 5, 7, to compare their standard deviations. The video suggests that the second set has a higher standard deviation because its numbers are more spread out.

  • 00:05:00 - 00:10:21

    The video continues with a detailed calculation of the standard deviation for the numbers 3, 5, 7, using the population standard deviation formula. The mean is calculated first, which is 5. The differences from the mean are calculated (3-5, 5-5, 7-5), squared, summed, and divided by the number of data points, followed by taking the square root to find the standard deviation, approximately 1.63. The same process is repeated for the numbers 4, 5, 6, with a calculated standard deviation of approximately 0.816, confirming it's lower due to less spread. The variance is addressed next, noted to be the square of the standard deviation, exemplified by squaring 1.63 to obtain approximately 2.66. The video concludes by summarizing that the methods for population and sample standard deviations mainly differ in the divisor used (n or n-1) and briefly discusses variance calculation.

マインドマップ

Mind Map

よくある質問

  • What is the formula for population standard deviation?

    The formula for population standard deviation is the square root of the sum of squared differences between each data point and the population mean, divided by the number of data points.

  • What is the difference between population and sample standard deviation formulas?

    The population standard deviation uses the actual number of data points, while the sample standard deviation divides by the number of data points minus one.

  • How do you find the mean of a data set?

    The mean is found by adding all the numbers in the data set and dividing by the number of data points.

  • Why does standard deviation measure dispersion?

    Standard deviation measures how spread out the numbers in a data set are, indicating the degree of variation from the mean.

  • What is the importance of understanding standard deviation?

    Standard deviation provides insight into the variability and consistency within a data set, making it useful for assessing risk and making comparisons.

  • Can you calculate variance from the standard deviation?

    Yes, variance is the standard deviation squared.

  • What is an example of calculating standard deviation in the video?

    The video shows the calculation of standard deviation for two sets of numbers: 3, 5, and 7, and 4, 5, and 6, using the population standard deviation formula.

  • What kind of subjects does the video creator offer?

    The video creator offers subjects in algebra, trade, pre-calculus, calculus, chemistry, and physics.

  • What is the approximate standard deviation for numbers 3, 5, and 7?

    The approximate standard deviation for numbers 3, 5, and 7 is 1.63.

  • How is variance calculated in the video?

    Variance is calculated by squaring the standard deviation, which is the sum of squared differences from the mean divided by the number of data points.

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  • 00:00:01
    in this video we're going to calculate
  • 00:00:03
    the standard deviation of a set of
  • 00:00:05
    numbers
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    now there's two formulas you need to be
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    aware of the first one
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    is the population
  • 00:00:11
    standard deviation
  • 00:00:17
    now this formula
  • 00:00:19
    is represented by the letter sigma
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    that's the standard deviation
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    it's equal to the sum
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    of all the differences between
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    every point
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    in the data set and the population mean
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    the population mean is mu
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    which is this symbol here
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    and then you need to square it
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    divided by
  • 00:00:45
    n which is
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    all of the
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    numbers in the set
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    and then you got to take the square root
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    of the whole result
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    so that's the population standard
  • 00:00:54
    deviation
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    the next formula is the sample standard
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    deviation
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    so let's say if
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    you have
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    just a sample of a population not the
  • 00:01:07
    entire population
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    if you just have a sample data out of
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    the entire data
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    then you want to use this formula
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    s which is the standard deviation
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    is equal to sigma
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    the sum of all of the
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    differences between every point
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    and the mean that's the sample mean
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    in the other equation we had the
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    population mean represented by mu
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    but this is the sample mean which is
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    basically the average of all the data
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    points in the set
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    and then you have to square it but it's
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    going to be divided by n
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    minus 1 as opposed to n
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    and so that's how you calculate the
  • 00:01:50
    standard deviation of the sample
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    now let's work on an example
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    let's say if we have two set of numbers
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    four five and six
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    and also three
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    five and seven
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    which one has a greater standard
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    deviation let's use the population
  • 00:02:11
    standard deviation formula
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    but if we had to guess
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    which set of numbers
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    has the greater standard deviation is it
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    the one on the left or the one on the
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    right
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    what would you say
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    we need to understand the basic idea
  • 00:02:26
    of standard deviation you need to know
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    what it measures
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    standard deviation tells you
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    how far apart the numbers
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    are related to each other
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    so the more spread out they are the
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    greater the standard deviation
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    four five and six are closer to each
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    other than three five and seven
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    and you could tell if you plot them on a
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    number line
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    let's put five in the middle
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    so 4 5 and 6
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    here they are on a number line
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    now in contrast
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    let's put the same numbers
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    on this number line
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    we're going to have 3
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    5 and 7.
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    so if you look at the the red points
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    the points in red are further apart
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    the points in blue they're very close
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    together
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    so therefore
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    four five and six
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    has a lower standard deviation in three
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    five and seven
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    so sigma is low
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    and here the sigma value is high
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    now go ahead and calculate the
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    standard deviation
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    for this
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    for the set of numbers three five and
  • 00:03:54
    seven
  • 00:03:55
    so what's the first thing that we should
  • 00:03:57
    do
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    the first thing that we should do is
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    calculate the mean
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    to find the mean
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    it's going to be the sum
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    of all the numbers divided by 3.
  • 00:04:09
    now because the three numbers are evenly
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    spaced apart the mean is going gonna be
  • 00:04:13
    the middle number five
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    three plus five is eight eight plus
  • 00:04:17
    seven
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    is fifteen
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    fifteen divided by three is five so
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    that's the mean
  • 00:04:27
    now what should we do next now that we
  • 00:04:28
    have the mean
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    now think of the formula
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    it's going to be a sigma
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    of every point minus the
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    mean squared
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    divided by
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    n
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    and then all of this is within the
  • 00:04:47
    square root
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    so here's how to use the equation first
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    we're going to use the first point 3
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    subtract it by the mean and then squared
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    next we're going to take the second
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    point 5
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    subtract it from the mean squared
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    and then it's going to be 7
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    minus 5
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    squared
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    so each of these three points
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    you're going to plug into x sub i
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    and then you're going to square the
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    differences between each of those values
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    and the sigma represents sum so you're
  • 00:05:19
    going to add every difference
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    that you get
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    or you can add the square of every
  • 00:05:25
    difference that you get
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    and now let's divide it by n
  • 00:05:29
    so n is the number of numbers that we
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    have in this set there are three numbers
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    inside
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    so n is 3
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    and then we're going to take the square
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    root of the entire thing
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    three minus five is negative two
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    negative two squared is four
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    five minus five is zero
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    seven minus five is two two squared is
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    four
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    four plus four is eight
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    so we have the square root of eight
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    divided by three
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    and at this point we're going to use the
  • 00:06:00
    calculator
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    8 divided by 3 is about 2.67 and if you
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    take the square root of that
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    you're going to get 1.63
  • 00:06:14
    so that's the standard deviation
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    for 3 5 and 7.
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    now let's calculate the standard
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    deviation
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    for the other set of numbers four five
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    and six so why don't you go ahead and
  • 00:06:28
    pause the video
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    and try this example
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    calculate the standard deviation using
  • 00:06:33
    the same formula
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    so let's go ahead and begin let's
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    calculate the population mean
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    it's going to be 4 plus 5 plus 6 divided
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    by
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    the number of numbers that we have which
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    is
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    3. four plus six is ten ten plus five is
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    fifteen and we know that fifteen divided
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    by three is five
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    so once again any time the numbers are
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    evenly spread apart the mean is going to
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    be the middle number
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    so now
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    we can calculate the standard deviation
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    so sigma
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    is going to equal the square root but
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    before we do that let's calculate
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    the differences
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    so the first difference that we have the
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    first number is going to be 4
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    and we're going to subtract it from the
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    mean
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    and then square it the next number
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    is 5
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    subtract it from the mean
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    and then square it
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    and then after that the last number is
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    six
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    this is going to be
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    six minus five squared
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    now it's divided by n
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    and let's not forget to take the square
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    root of the entire thing
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    four minus five is negative one negative
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    one squared is simply one
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    five minus five is zero
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    six minus five is one
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    and it's all divided by three one plus
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    one is two
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    so we have the square root of two
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    divided by 3.
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    now 2 divided by 3 as a decimal is about
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    0.67
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    and the square root of 0.67
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    is 0.816
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    so as you can see the standard deviation
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    is less because
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    these numbers are closer to each other
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    they're not far apart from the mean
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    in the other example
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    three five and seven
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    they're further apart from the mean
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    which is five
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    three is two units away from five
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    four
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    is only one unit away from five and
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    that's why the standard deviation is so
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    much less
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    now let's go back to the first example
  • 00:08:39
    we said that the population standard
  • 00:08:41
    deviation
  • 00:08:42
    is approximately 1.63
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    so given this information
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    how can you calculate
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    the variance
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    v a r i a-n-c-e
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    how can we find the variance
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    the variance is simply the square
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    of the standard deviation
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    so 1.63 squared
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    is equal to now keep in mind this is a
  • 00:09:09
    rounded answer
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    i don't remember what the exact answer
  • 00:09:12
    was but
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    once you square it it's about 2.66
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    so that's how you can calculate the
  • 00:09:18
    variance
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    the formula for variance is basically
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    the sum
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    of all the square differences
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    between every point and the population
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    mean
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    divided by n
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    it's basically the same formula without
  • 00:09:32
    the square root symbol
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    well that's it for this video so now you
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    know how to calculate
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    the population standard deviation and
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    also the sample standard deviation even
  • 00:09:44
    though we did just one of them the
  • 00:09:46
    process is the same
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    of finding the other one
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    the only difference is you have n minus
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    one instead of n
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    you also know how to calculate the
  • 00:09:55
    variance as well
  • 00:09:57
    so that concludes this video by the way
  • 00:10:00
    if you want to find more of my videos
  • 00:10:01
    you can check out my channel or visit my
  • 00:10:03
    website video.tutor.net and you can find
  • 00:10:06
    playlists on algebra trade pre-calculus
  • 00:10:09
    calculus
  • 00:10:11
    chemistry and physics
  • 00:10:13
    so those are the subjects that i
  • 00:10:14
    currently offer right now
  • 00:10:16
    and if you're interested just feel free
  • 00:10:18
    to check that out
  • 00:10:19
    so thanks again for watching
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