**Understanding the Normal (or Gaussian) Distribution**

The normal distribution is often referred to as a bell curve, simply because it resembles a bell.

Characteristics

The Normal Distribution has:

Now Let's Take a Look at the Standard Deviation of a Normal Distribution...

First off, what is standard deviation?

The normal distribution (also known as the Gaussian distribution) is a special probability model where the data is scattered symmetrically about a central, mean value.

Some Real World Examples:

Heights of People

IQ Scores

Grades on a Test

Most males in America would be around 5' 8'' tall and would lie in the middle of the distribution.

To calculate standard deviation, you can use the following formula:

Given that you have data that follows the normal distribution, when you calculate the standard deviation of this data:

Using our knowledge of standard deviation, how can we apply it to our measurements?

Being able to calculate standard deviation allows us to decide whether two measurements are statistically similar or statistically divergent.

When comparing two measurements A and B...

For instance, if you were to measure the mass of a spoonful of sugar a thousand times, the data points would produce a histogram following a normal distribution.

But what if you specifically wanted to calculate standard deviation of the mean (like in the picture we saw in the slide before)? It turns out that there's a formula for this as well:

- 1

Where Does the Normal Distribution Come From?

Imagine you measure the mass of a spoonful of sugar 10 times and obtain the following values in grams:

43.77, 42.66, 42.63, 42.14, 42.06, 40.56, 39.90, 39.83, 40.02, 40.09

From these values, we can calculate the mean to be 41.37g and the standard deviation to be 1.44 g.

Therefore if you measure another spoonful of sugar and obtain 39.00 g, you can state with

68% confidence

that the mass of sugar is in the range:

39.00 +/- 1.44 g

In particular,

standard deviation of the mean:

The equation for standard deviation applies to "any given measurement."

Standard Deviation of the MEAN

Standard deviation is an estimate of the average uncertainty in measurements.

43.77, 42.66, 42.63, 42.14, 42.06, 40.56, 39.90, 39.83, 40.02, 40.09

Given the same 10 measurements of a spoonful of sugar in grams:

When you calculate standard deviation of the mean (where the mean is 41.37 g) you obtain 0.456 g.

Therefore, you may state:

with 68% confidence that the average mass of a spoonful of sugar is within 1 standard deviation:

41.37 +/- 0.456 g

with 95% confidence:

41.37 +/- 0.912 g

with 99.7% confidence:

41.37 +/- 1.368 g

DISCLAIMER:

You typically need more than 10 data points to establish a normal distribution. In fact, it often takes a few hundred points to establish a normal distribution. Although for the sake of easy calculation examples, we stuck to a small sample size.

The normal distribution occurs due to random error. Random errors in measurement are equally likely to push our readings above or below the true value.

a mean, median, and mode that are all equal to each other

(remember: the mean is the average of all the values, the median is the middle value, and the mode is the number that appears most frequently)

symmetry about the center

50% of the values below the mean and 50% of the values above the mean

Those taller would be found on the right, and those shorter would be found on the left.

As we approach the right side of the curve, the

height of an individual

gradually increases on the

x-axis

. Meanwhile, the

number of individuals

gradually decreases on the

y-axis

.

If we were to go back to the histogram of grades on a test... we'll see that each of the colored bars is equivalent to one standard deviation.

68% lie within ONE standard deviation of any given measurement

95% lie within TWO standard deviations of any given measurement

99.7% lie within THREE standard deviations of any given measurement

(http://icons.webtoolhub.com/icon-n13441f32431-detail.aspx)

(http://www.mathsisfun.com/data/standard-normal-distribution.html)

(http://thegatewayonline.ca/article/view/curve_your_enthusiasm)

(http://www.free-iqtest.net/iq-score-guide.asp)

(http://www.stat.sfu.ca/features/histogram.html)

(http://physicsofhope.blogspot.com/)

(http://commons.wikimedia.org/wiki/File:Black_coffee_with_sugar_-_Evan_Swigart.jpg)

(http://www.mathsisfun.com/data/standard-normal-distribution.html)

(http://jamesstacks.com/stat/norm_areas_files/image001.jpg)