NORMAL-DISTRIBUTION ( Statistics and Probability)

Statistics and probability

1. lesson 4: normal distribution

2. NORMAL DISTRIBUTION The normal distribution, or simply as normal curve is a bell-shaped distribution that has an important role in inferential statistics. It provides a graphical representation of a statistical values that are needed in describing the characteristics of a population as well as in making decisions.

3. NORMAL CURVE

4. PROPERTIES OF NORMAL CURVE 1. A normal probability distribution is bell-shaped.

5. PROPERTIES OF NORMAL CURVE 2.curve is symmetrical to its center.  SYMMETRICAL SIDE

6. PROPERTIES OF NORMAL CURVE 3.the mean, median, and mode coincide at the center. MEAN = MEDIAN = MODE

7. PROPERTIES OF NORMAL CURVE 4.tHE TAILS OF THE CURVE FLATTEN OUT IN THE X AXIS BUT NEVER CROSSES . ASYMPTOTIC TAIL

8. PROPERTIES OF NORMAL CURVE 5.tHE AREA UNDER THE CURVE IS 1 OR 100%.

9. AREAS UNDER THE NORMAL CURVE 50% 50%

10. AREAS UNDER THE NORMAL CURVE Z SCORES

11. AREAS UNDER THE NORMAL CURVE Z = -1 TO Z = 1 68.26%

12. AREAS UNDER THE NORMAL CURVE Z = -2 TO Z = 2 95.44%

13. AREAS UNDER THE NORMAL CURVE Z = -3 TO Z = 3 99.74%

14. Z TABLE is based on a normal distribution with mean µ = 0 and standard deviation σ = 1 called standard normal distribution.

15. FINDING THE AREA UNDER THE NORMAL CURVE

16. LET'S TRY! 1. Find the area that corresponds to Z = 1 z = 1 corresponds to the area 0.3413 or 34.13%

17. LET'S TRY! 2. Find the area that corresponds to Z = 1.05 Z = 1.05 corresponds to the area 0.3531 or 35.31%

18. LET'S TRY! 3. Find the area that corresponds to Z = -1.05 z = -1.05 corresponds to the area 0.3531 or 35.31% -1.05

19. LET'S TRY! 4. Find the area that corresponds to Z = 2.38 z = 2.38 corresponds to the area 0.4913r 49.13%

20. LET'S TRY! 5. Find the area that corresponds to Z = -2.16 z = -2.16 corresponds to the area 0.4846r 48.46%

21. LET'S TRY! 6. Find the proportion of the area between  z = 1 and Z = -1 Between a and b Sign Notation Operation Same Sign P (a < z < b) Subtract the biggest area by the smallest area Different Sign P (- a < z < b ) P (a < z < - b ) Add the biggest area by the smallest area z = 1, A = 0.3413 z = -1, A = 0.3413 P(-1<z<1) = 0.3413 + 0.3413     = 0.6826 or 68.26% Therefore, P(-1<z<1 ) is corresponds to the area 0.6826 or 68.26%

22. LET'S TRY! 7. Find the proportion of the area between  z = 2  and Z = -1.5 Between a and b Sign Notation Operation Same Sign P (a < z < b) Subtract the biggest area by the smallest area Different Sign P (- a < z < b ) P (a < z < - b ) Add the biggest area by the smallest area z = 2, A = 0.4772 z = -1.5, A = 0.4332 P(-1.5<z<2) = 0.4772 + 0.4332     = 0.9104 or 91.04% Therefore, P(-1.5<z<2 ) is corresponds to the area 0.9104 or 91.04%

23. LET'S TRY! 8. Find the proportion of the area between  z = 2  and Z = 1 Between a and b Sign Notation Operation Same Sign P (a < z < b) Subtract the biggest area by the smallest area Different Sign P (- a < z < b ) P (a < z < - b ) Add the biggest area by the smallest area z = 2, A = 0.4772 z = 1, A = 0.3413 P(1<z<2) = 0.4772 - 0.3413     = 0.1359 or 13.59% Therefore, P(1<z<2 ) is corresponds to the area 0.1359 or 13.59%