We hear the word “normal” all the time, referring to everything from our homes to our health.  We use the word carelessly, loosely referring to anything that seems the same as any other, such as a “normal” house, on a “normal” street, in a “normal” town.  It seems to imply that they are very much like most others...in other words they are very average.  In fact, that is what the word normal means, but there’s a little bit more to it.

In mathematics—or more specifically, statistics—the word normal refers to a type of probability distribution: a way to calculate the chances that a specific event will occur using mathematical equations (click here for more information about probability).  It also describes the way these events look on a graph.  Here is a simple example:  imagine a 10 mile stretch of a very busy road.  All 10 miles of road was constructed at the same time and is therefore the same age, and gets the same wear.  The engineers who built this road know that at some point in the next year, the road will develop a crack. 

For our purposes, we assume that the probability of the crack forming is the same at any point in the road.  Let’s look at what this looks like if we turn this information into a graph, with the probability of a crack forming on the vertical axis and miles on the horizontal axis:

We can see this graph is just a horizontal line, representing the equal probability of the crack forming at any point along the 10 mile stretch of road.  Because the probability is uniform across the whole range of possible values—that is, along the whole stretch of road that we are considering—we call this a uniform distribution. 

The normal distribution is also a probability distribution, but in this case not all of the possible outcomes have equal probability.  In the normal distribution, there is one outcome value which is the average of all the possible values, and this value has the highest probability.  All values higher and lower than this value have lower and lower probabilities the farther they are from the average.  For example, think about the height of men in the United States.  McDowell reports that the average height for men over 20 years old in the US is 5 feet, 9.5 inches (1).  However, we know there are men that are shorter and men that are taller: currently the tallest man in the world is 8 feet, 3 inches tall (2); the shortest man in the world is only 1 foot, 9.5 inches tall (3)!  The probability of anyone being this tall (or this short) is very, very low.  If we could measure the height of 1000 men from all over the US, and we could plot this information on a graph with number of men on the vertical axis and height measured (in feet) on the horizontal axis, it would look like this:

The peak of this curve occurs at a mean (that is, at an average) of 5.79 feet which is 5 feet, 9.5 inches—the average height for men over 20 years old in the US.  This means men from our sample have a high probability of being around this height.  As the heights get taller or shorter, the probability of finding a man in our sample of this height goes down.  This makes the curve look like a bell, which is why it is often called the “bell curve”. So when we say something is “normal”, it really does mean that it has a high probability of being similar to the average!

TRY THIS!

Here’s what you’ll need:

1)      A pen or pencil

2)      A piece of paper

3)      A computer with a spreadsheet program installed, such as Microsoft Excel®

4)      At least 30 individuals of around the same age (50 is even better).  If you are in school now, you can use your classmates!

5)      A tape measure at least 10 feet long.

Here’s what you need to do:

1)       Make a table on your piece of paper with height (in feet) in one column, and number of individuals in another.  In the height column, write numbers ranging from 3-7 in increments of 0.5, as in the following example:

2)      Ask each of the individuals of about the same age how tall they are.  If they are not sure, measure them with the tape measure!

3)      As each individual tells you their height, make a mark in the other column across from the height range they fall into.  Here is some example data:

4)      When you are finished, open your spreadsheet program and make one column with your numbers for how many people were in each height range.

5)      Use these numbers to make a column chart.  In Excel, you can do this by selecting the data you wish to graph, and selecting the insert tab.  Select the column chart under the ‘chart’ options, and choose the type on the top left corner. 

When we use our sample numbers, this is what our graph looks like:

Notice it looks a lot like the graph of the normal distribution we saw earlier. 

What does your chart look like?  If you connect the tops of all the columns, what shape does it look like?  Is the highest point roughly in the middle?  Print out this graph and save it with your notes!

CHALLENGE YOURSELF!

What else do you think might have a normal distribution?  Try measuring the weight of each nail in a box, or count the number of seeds in a bunch of identical seed packages.  Be sure to use at least 30 nails or seed packages, or more if you can; the more you use for your testing (that is, the larger your sample size) the better you will be able to see the probability distribution.  Be sure you keep track of you numbers the same way as before, and use a spreadsheet program to graph all the numbers (weight or number of seeds on the horizontal axis, and number of observations within a certain range on the vertical axis).  This way you can get an idea of what the probability distribution might look like. 

References:

1)      McDowell, Margaret A. et al. (October 22, 2008). "Anthropometric Reference Data for Children and Adults: United States, 2003–2006". National Health Statistics Reports, 10.

2)      “World’s Tallest Man—Living”. Guinness Book of World Records .  Accessed 8/21/14.  Last updated 2/8/2011.

3)      Sheridan, Michael. (February 26, 2012).  “Chandra Bahadur Dangi is world’s shortest living man: Guinness World Records.”  New York Daily News.  www.nydailynews.com.

When someone says, “What are the chances of that happening?” what are they asking?  Usually when you hear this, it means ‘that’ is very unlikely (as in, “What are the chances that Santa will bring me a pony for Christmas?”).  However, this question really has a deeper meaning.  Let’s explore what this question really means by focusing on the key word: chances.

When we talk about chances, we are really talking about something called probability.  Probability is just how likely an event is to occur.  So if something has a high probability, this means it is very likely to happen (as in, what is the probability your mom will make you brush your teeth before bed?).  If something has a low probability, it means it is not very likely to happen (as in, what is the probability your mom will let you eat cupcakes for dinner?).  Although in reality, it is not always so obvious how likely or unlikely an event is to happen!

So how do we decide how likely or unlikely an event is to happen?  More importantly, can we predict how often this event will happen?  As it turns out, we can!  If we know how many different outcomes there are for a certain event, we can figure out just how likely each one really is. 

Here’s an example:  take a normal die (this is the singular of dice...so if you have a pair of dice, just use one for this!).  A normal die has six sides, each with a different number 1 through 6. 

If we roll this die, there are six possible numbers we could get, so there are six possible outcomes, and each one is equally likely to happen; one of these outcomes is rolling a number 3.  To find the probability of rolling a number 3, we first count how many times the number 3 appears on the die—we know the number 3 appears only one time on this die.  Now we count how many possible outcomes there are—we know there are six different possible outcomes, since there are six sides on the die, and each one is equally likely to happen.  To find the probability of rolling the number 3, we take the number of times this outcome appears on the die (it appears one time), and divide this number by the total number of possible outcomes (six).

When we divide 1 by 6, we get 0.167 (rounded to three decimal points).  What does this mean?  Probability may always be expressed as a decimal number between 0 and 1, with 0 meaning the event will never happen, and 1 meaning the event will always happen.  This event is closer to 0 than to 1, meaning it is not very likely, but it is not impossible either.  Probabilities can also be expressed as a percent (%), with 0 being 0%, and 1 being 100% (just multiply the decimal number by 100 to get the percent chance of this event happening).  Rolling a number 3 on our six-sided die therefore has a probability of 16.7% (0.167 X 100 = 16.7%). 

An important thing to note about our die is that each of the numbers has an equal chance of coming up on a roll.  Since each number only appears one time, we know that each of these numbers has an equal probability of coming up.  So the probability of each number is the same as for the number 3, or 16.7%.  If we add up the probability of each of the numbers appearing (that is, 16.7% + 16.7% + 16.7% + 16.7% + 16.7% + 16.7%--we have six numbers, all with the same probability, so we add 16.7% six times), we should get a number that is just about 100%.  16.7% + 16.7% + 16.7% + 16.7% + 16.7% + 16.7% = 100.2%; this extra 0.2% is because we rounded to three decimal places—this number is close enough to 100% that we know we calculated correctly.  When you calculate probabilities, remember:  IF YOU ADD UP THE PROBABILITIES OF ALL THE POSSIBLE EVENTS, THEY SHOULD EQUAL 100%!

TRY THIS!

Here’s what you’ll need:

1.       A small bag of Skittles® candies

2.       A small paper bag

3.       A piece of paper

4.       A pen or pencil

Here’s what to do:

1.      Pour 10 Skittles out of your bag onto the table.  DON’T PICK OUT THE COLORS YOU WANT; JUST MAKE SURE THERE ARE 10 SKITTLES!

2.       Write on your piece of paper the names of the different colors in your sample: Red, Orange, Yellow, Green, and Purple, and divide these with long, vertical lines.  Beside the names of the colors, write down the number of the Skittles of that color you have in your sample.  For example, if you drew 3 red Skittles, write Red 3.

3.       Under the names and numbers for each color, write the number of Skittles of that color, divided by 10 (the total number of Skittles).

4.       Place all 10 Skittles in the paper bag.
5.       Calculate based on these numbers the probability of each color being drawn from the paper bag at random (that is, without looking or actively choosing the color).  For example, for 3 red Skittles, 3/10 should be written, and this works out to 0.3 or 30%.  So we know there is a 30% chance of a red Skittle being chosen at random from this sample.

6.       Add up all the percentages.  They should equal 100%, or very close. 

7.       Reach into the paper bag containing the 10 Skittles, and pull one out.  Look at the color, and make a tick mark under that color on your sheet of paper.  Put the Skittle back into the bag, and shake the bag to mix the Skittles well.

8.       Repeat this process 99 times, for a total of 100 tick marks.  Be sure to put the Skittle back in the bag each time!

9.       Count the number of times each color was randomly selected from the bag.   Calculate the percent of the time this color was randomly selected by dividing this number by 100.  Compare this number to the percentage you calculated earlier. 

What have you found?  How close was the percentage you calculated to the number of times out of 100 that each color was chosen?  If any of the percentages were very different, why might that be?

WHAT ELSE CAN YOU TRY?

·         For this experiment, you pulled a Skittle out of the bag 100 times.  That’s a lot, but the more times you repeat your experiment, the more accurate the number will be.  Try repeating this experiment 1,000 times!  See if the actual numbers get any closer to your predictions!

·         You can also try the same experiment with a new sample of Skittles.  Pick out 20 different Skittles this time and instead of dividing by 10 to get your probability divide by 20!

·         Try this with something different:   Use a different candy, or a deck of cards!


Skittles® are a trademark of the Wm. Wrigley Jr. Company.