Bell Curve

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Ask Not For Whom the Bell Curves...

DEFINITIONS

Data are observations that have been collected.
Measured Data are a set of numbers collected by measuring our observations.
A Frequency Distribution is a way of organizing data to reveal patterns.
The data can be more easily seen by grouping them together in intervals.
A Histogram or Bar Chart is a picture of these data intervals.
A Bell Curve or Normal Curve results when the bars in a histogram are extremely thin
and data are distributed normally.
A Pie Chart is a circular way of presenting data frequencies.
3 Measures of Central Tendency
1. The Mean, or Average, of a set of numbers is their sum divided by their set size.
2. The Median is the number in the middle when the numbers are arranged in order of size.
3. The Mode is the number that occurs most frequently, if there is such a number.
3 Measures of Variability
1. The Range of a set of numbers is the difference between the largest and the smallest numbers in the set.
2. The Standard Deviation of a set of numbers is determined by finding:
  1. The mean of the numbers,
  2. The difference between each number in the set and the mean,
  3. The squares of these differences,
  4. The mean of the squares, and
  5. The square root of this mean.
3. The Variance is the square of the Standard Deviation.
A Normal Distribution is a set of numbers where the Mean, Median, and Mode are all the same.
In a Normal Distribution of numbers:
1. 68% of the numbers are within one standard deviation of the mean,
2. 96% are within two standard deviations of the mean, and nearly
3. 100% are within three standard deviations of the mean.
A Sample is a group of items chosen to represent a larger group called the Population.
A Representative Sample is a sufficiently large and random sample.
A Random Sample is a subset of the population where each item in the sample has an equal chance of being included.

Binomial Distribution

BINOMIAL DISTRIBUTION

The mathlet above displays the probability distribution of any number of independent binary events. Input the number of trials, the chance for each success, the height of the tallest bar graph, and the maximum and minimum for the range.

Initially, the mathlet displays the probabilities of flipping x number of heads in 10 tosses; and, also, the probabilities of x number of boys in a family with 10 children. Remember, if a distribution is binomial, then there are only two outcomes: heads or tails, boy or girl, yes or no, true or false, right or wrong, win or lose. Each probability is the complement of the other. Also each trial run is independent of all the other trials. Many of our coin, card, and dice experiments were examples of binomial probability.

For our ESP game, input 5 for the number n of trials (number of cards). Input .25 for the chance p of guessing a suit. Input 1200 for the height of the tallest bar. Input 5 for the maximum. The numbers on the bottom ( 0 to 5) represent the number of possible correct guesses. The number on top of each bar is the probability of correctly guessing the number on the bottom.

For our 4 dice game, input 4 for the number n of trials (number of dice). Input .1667 for the chance p=1/6 of rolling a six on one die. Input 900 for the height of the tallest bar. Input 4 for the maximum. The numbers on the bottom ( 0 to 4) are the probabilities of rolling 0, 1, 2, 3, and 4 sixes, respectively. The number on top of each bar is the probability of correctly guessing the number on the bottom.

When measuring the probability distribution of binary events, the Variance can be approximated as the product of the number of trials, the probability of a win, and the probability of a loss: VAR = npq. The square root of the Variance is the Standard Deviation. Likewise, the Mean, or Average, can be approximated as the product of the number of trials and the probability of a win: AVG = np.

FOR DISCUSSION

To toss 100 coins:
1. Input 100 for Trials
2. Input .5 for P(win)
3. Input 6500 for Height
4. Input 100 for Maximum
5. Press the ENTER key
6. What do you see?
7. What do you not see?
In the upper left corner of the mathlet are the expected Average and Standard Deviation for our coin toss.
How many Heads should we expect to toss on average in 100 flips?
What is the expected Standard Deviation?
What is the meaning of the Standard Deviation?
How many "stair steps" can you count in the graph?
The Standard Deviation tells us how many of those steps equal 1 Standard Deviation.
How many standard deviations are visible on the graph?
Answer: divide the number of red steps by the Standard Deviation!
What is the range of visible red steps?
What number of coins is represented by the shortest red step on the left?
Enter this number in Minimum box.
What number of coins is represented by the shortest red step on the right?
Enter this number in Maximum box. Press ENTER key and zoom in on the curve.
Subtract the Standard Deviation from the Average.
This number is -1 Standard Deviations from the Average.
Add the Standard Deviation from the Average.
This number is +1 Standard Deviations from the Average.
What is sum of the probabilities between -1 and +1 Standard Deviations?
What is sum of the probabilities between -2 and +2 Standard Deviations?
What is sum of the probabilities between -3 and +3 Standard Deviations?
Compare these sums to the probabilities in the Bell Curve at the top of the page?
What are the differences between them?
Why do you think there are differences between our bar chart and the Bell Curve?
Increase the number of coins to 1000.
1. Enter 1000 in the Trials box
2. Enter 2000 in the Height box
3. Enter 0 in the Minimum box
4. Enter 1000 in the Maximum box
5. Press the ENTER key.
What happens to the shape of our coin-tossing bar chart as the number of coins increases?
How many coins would we need to toss before the probabilities in our bar chart equal those in the Bell Curve?

Coin Toss

COIN TOSS

QUESTIONS

Input 10 for Coins
Input 1000 for Tosses
Input 0 for Minimum
Input 10 for Maximum
Press the ENTER key
Run this coin toss 10 times
Write down the Mean number of HEADS each of the 10 times
Write down the Standard Deviations each of the 10 times
Make a table as below
AVERAGE each column — you can use the Average mathlet below,
or simply add all 10 entries and divide by 10
Use the Binomial Bar Chart to calculate the BINOMIAL numbers:
1. How many times should 5 heads turn up?
  Write this number on the BINOMIAL row in the MEAN column
2. How many times should 4, 5, or 6 HEADS turn up?
  Write this number on the BINOMIAL row in the ±1 STD column
3. How many times should 2, 3, 7, or 8 HEADS turn up?
  Write this number on the BINOMIAL row in the ±2 STD column
4. How many times should 1 or 9 HEADS turn up?
  Write this number on the BINOMIAL row in the ±3 STD column
5. How many times should 0 or 10 HEADS turn up?
  Write this number on the BINOMIAL row in the ±4 STD column
Write the Expected number on the BINOMIAL row
Subtract the Expected numbers from the Average numbers
Write the absolute differences on the DIFFERS row
Use the Normal Bell Curve to calculate the NORMAL numbers:
1. You should have tossed 4, 5 or 6 heads about 68% of the time.
  Write this percentage on the NORMAL% row in the ±1 STD column.
  On average, what percentage of the time did you toss 4, 5 or 6 heads?
  Write this percentage on the AVERAGE% row in the ±1 STD column.
2. You should have tossed 2, 3, 7 or 8 heads about 27% of the time.
  Write this percentage on the NORMAL% row in the ±2 STD column.
  On average, what percentage of the time did you toss 2, 3, 7 or 8 heads?
  Write this percentage on the AVERAGE% row in the ±2 STD column.
3. You should have tossed 1 or 9 heads about 4% of the time.
  Write this percentage on the NORMAL% row in the ±3 STD column.
  On average, what percentage of the time did you toss 1 or 9 heads?
  Write this percentage on the AVERAGE% row in the ±3 STD column.
4. You should have tossed 0 or 10 heads about 1% of the time.
  Write this percentage on the NORMAL% row in the ±4 STD column.
  On average, what percentage of the time did you toss 0 or 10 heads?
  Write this percentage on the AVERAGE% row in the ±4 STD column.
What could we do in our experiment to more closely approximate the normal bell curve?

TOSS	MEAN	±1 STD	±(1-2) STD	±(2-3) STD	±(3-4) STD
1
2
3
4
5
6
7
8
9
10
AVERAGE
BINOMIAL
DIFFERS
AVERAGE%
NORMAL %

Comments:

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Last Modified 11/11/08 6:44 PM