Normal curve

introduction of normal curve

1. Normal Curve

2. HISTORY Began in the middle of 18 th century with the work of Abraham DeMoivre Marquis de Laplace At the beginning of 19 th century Karl Friedrich made substantial contributions. And scientist called it the “Laplace-Gaussian Curve”. Karl Pearson is credited with being first to refer to the curve as the “Normal curve

3. IMPORTANCE One reason the normal distribution is important is that many psychological and educational variables are distributed approximately normally. -Measures of reading ability, introversion , job satisfaction, and memory are among the many psychological variables approximately normally distributed . -Although the distributions are only approximately normal , they are usually quite close.

4. IMPORTANCE The second reason the normal distribution is so important is that it is easy for mathematical statisticians to work with. -This means that many kinds of statistical tests can be derived for normal distributions. -Almost all statistical tests discussed in this text assume normal distributions. -Fortunately , these tests work very well even if the distribution is only approximately normally distributed . -Some tests work well even with very wide deviations from normality.

5. CHARACTERISTIC Asymptotic : Approaching the X-axis but never touches it. Symmetric : made up of exactly similar parts facing each other Symmetric Asymptotic X

6. CHARACTERISTIC Ranges from negative infinity to positive infinity. With two tails tails

7. WHAT IS THE NORMAL CURVE This is when the data is distributed evenly around a middle value. The data is symmetrical about the middle value .

8. WHAT IS THE NORMAL CURVE This is also called a “BELL CURVE” because it looks like a bell.

9. WHAT IS THE NORMAL CURVE Half the data falls above and half below the middle value. 50% 50%

10. MEAN – MEDIAN - MODE The mean is the average of all the data in the distribution. The median is the middle value of the data ordered from smallest to largest The mode is the value that occurs most often in the data. In a normal distribution , the mean , median and mode are the same or equal or identical

11. STANDARD DEVIATION is how the spread out the numbers are from the middle value. Data is said to fall within a specific number of standard deviations when it is not the middle value. -1 +1 -2 +2 -3 +3 A normal distribution follows the 68-95-99.7 rule.

12. 68-95-99.7 RULE also called the Empirical Rule 68% of the data falls within 1 standard deviation of the middle value. 95% of the data falls within 2 standard deviations of the middle value. 97% of the data falls within 3 standard deviations of the middle value.

13. PERCENT AT EACH STANDARD DEVIATION The middle value represents 50%. Recall that 68% of the data falls within 1 standard deviation of the mean. Half the 68% falls above and half below 50%. 1 standard deviation below the mean is 50% - 34% = 16% 1 standard deviation above the mean is 50% + 34% = 84%

14. PERCENT AT EACH STANDARD DEVIATION Recall that 95% of the data falls within 2 standard deviations of the mean. Half the 95% falls above and half below 50%. 2 standard deviations below the mean is 50% - 47.5% = 2.5% 2 standard deviations above the mean is 50% + 47.5% = 97.5%

15. PERCENT AT EACH STANDARD DEVIATION Recall that 99.7% of the data falls within 3 standard deviations of the mean. Half the 99.7% falls above and half below 50%. 3 standard deviations below the mean is 50% - 49.85% = .15% 3 standard deviations above the mean is 50% + 49.85% = 99.85%

16. HOW TO USE NORMAL CURVE TO DETERMINE PROBABILITY Determine the mean. (µ) Determine the standard deviation. ( ơ ) Plot the mean and SD on the normal curve. Analyze the problem. Apply the 68-95-99.7 or empirical rule

17. Practice The normal distribution below has a standard deviation of 10. Approximately what area contained between 70 and 90? 40 60 70 80 90 100 50 Mean= 70 SD= 10

18. Practice For the normal distribution below, approximately what area contained between -2 and 1 1 -1 2 3 -3 -2 Mean = 0 SD= 1

19. Practice A certain variety of pine tree has a mean trunk diameter of µ=150cm and a standard deviation of ơ =30cm. A certain section of a forest has 500 of these trees. Approximately how many of these trees have a diameter of between 120 and 180 centimeters?

20. 34% + 34% = 68% 68% of 500 340 trees