Normal Distribution – Explained Simply (part 1)

I describe the standard normal distribution and its properties with respect to the percentage of observations within each standard deviation. I also make reference to two key statistical demarcation points (i.e., 1.96 and 2.58) and their relationship to the normal distribution. Finally, I mention two tests that can be used to test normal distributions for statistical significance.

normal distribution, normal probability distribution, standard normal distribution, normal distribution curve, bell shaped curve
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11 replies
  1. Brad Anderson
    Brad Anderson says:

    HERE's an idea.

    If each of you cut out a picture of this curve and throw 100 darts at its center, the numbers he has provided are guaranteed to be almost the exact same number of holes made in each section.

  2. Brad Anderson
    Brad Anderson says:

    Let me try to explain these percentages for the students that are struggling with it.

    The good news, is this: It is NOT within your ability to calculate these percentages on your own and you will not be expected to. This is a problem that can only be solved through advanced integral calculus, and it is problem that only a small handful of calculus enthusiasts have the ability to solve. BUT, you better be damn clear about what these numbers mean. Suppose this bell curve is a dance floor, and suppose there are exactly 10,000 people crowded asshole to elbow on this dance floor. Then, the middle two boxes will have exactly 3413 people, each. And, the outer two boxes will contain exactly 1359 people each. Add these totals and subtract them from 10000 and the difference will be split between the two outermost boxes. 10000-3413–3413-1359-1359 = 456.

    The best example of all is to think of this bell curve as a an upside down picture of Hoover Dam. Then, these numbers would represent the total weight of the hydrostatic pressure behind each section of its wall. The only real challenge you must face is being able to calculate the area of the independent sections by subtracting them from the cumulative sum of the distribution function.

    What you are calculating is the "area under a curve"; or how many yard of carpet it would take to cover this dance floor. Area, is most often and most easily calculated by the formula AREA = WIDTH x HEIGHT, but this assumes a RECTANGULAR shape. This formula does not work for a triangle until you realize that every rectangle is made of two equal triangles. Hence the modified equation AREA = 1/2 WIDTH x HEIGHT, which is still easy to calculate because the change in length of the sloped side is still a straight line. If we change the sloped side of this wall to a semi-circular, most math students can still calculate the area based on the area formula for a circle, but it is definitely getting more challenging. Next, if we changed the curved wall to a parabola (like x-squared) then the precise area can only be solved by advanced integral calculus. And the slope, although no longer static, can be calculated. BUT, 'e' none as euler's constant (2.7173…) is based on infinite layers of change that are difficult to pin down even with integral calculus. Finally, the equation for the bell curve e^(-x^2) requires the calculation of an integral that is almost impossible to calculate. THIS IS WHY YOUR STATISTIC BOOK HAS LOVINGLY DISPLAYED EVERY POSSIBLE VALUE OF THIS CURVE'S AREA IN ONE OF ITS APPENDICES. But, if you are hell bent on pulling these numbers out of your ass then this is the least challenging way to do it:::(1) SUB-Divide the chart into as many smaller boxes as you can, and calculate the area for each 'micro-box' the way you would for a rectangle. (2.) Add the area EACH of the micro-boxes, and it will be a very accurate approximation of the actual area under the curve. As the number of boxes in your calculation approaches an infinite number of infinitesimally small micro-boxes, your approximation will become more and more accurate. This dilemma should serve to make you aware of the magic of calculus, but it is not something you should be sweating over.

    Many of You are thinking of it backwards. 34.13% is the definition of what 1 standard deviation is. STANDARD DEVIATION is a method of describing how closely the data points adhere to the curve. From looking at the symmetrical shape of the bell curve, you can all tell that the average value (expected value) is smack dab in the center. But, we can stretch or squeeze this graph to infinite proportions without changing this central value. So, the next most important thing to calculate is the average distance each data point veers from dead center. The seemingly random fact that the average swerving distance of each data point from its center lane is + or – 34.13% is the feature which defines 1 standard deviation in this case scenario, and 1 standard deviation of + or – 34.13% is the key feature of the normal distribution curve.

  3. Carie5807 MSP
    Carie5807 MSP says:

    Well, I have a statistics test tomorrow 😢 hope this was able to sink in.. wish me luck

  4. VAUGHN CASTLE
    VAUGHN CASTLE says:

    OK. You totally lost me. Where are you getting theses number's from? How did  you get 34.13% ?so lost!

  5. farhat mohammadi
    farhat mohammadi says:

    I can't finish the video. The smacking of the lips is beyond annoying. Please refrain from doing that in future videos

  6. Slytherin Snowflake
    Slytherin Snowflake says:

    yep, still confusing the fuck out of me. WHERE DOES THE RANDOM 34.13% COME FROM?!?!?! god i hate math, everything about is unclear and unnecessary

  7. Douglas Maier
    Douglas Maier says:

    if you have a life example with a mean of 3 and SD of 1.5 how do you calculate probability when everything past the second negative variable is impossible.

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