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Normal Distribution

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Published in: Mathematics | Statistics
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Intro to Normal Distribution with a few examples to follow.

Masud H / Abu Dhabi

20 years of teaching experience

Qualification: Qualified Mathematics Teacher

Teaches: Mental Maths, Mathematics

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  1. ormal distribution 1. 2. 3. Learn about the properties of a normal distribution Solve problems using tables of the normal distribution Meet some other examples of continuous probability distributions
  2. Types of variables Discrete & Continuous Describe what types of data could be described as continuous random variable X=x — Arm lengths — Eye heights
  3. REMEMBER your box plot Middle ran
  4. The Normal Distribution (Bell Curve) Average contents 50 Mean = p = 50 Standard deviation = o — 5 O -30 -O O 30
  5. The normal distribution is a theoretical probability he area under the curve adds up to one
  6. A Normal distribution is a theoretical model of the whole population. It is perfectly symmetrical about the central value; the mean p represented by zero.
  7. he X axis is divided up into deviations from the mean. Below the shaded area is one deviation ro
  8. Two standard deviations from the mean
  9. Three standard deviations from the mean
  10. A handy estimate — known as the Imperial Rule for a set of normal data: I/o of data will fall within 10 of the I-I 0.683 68.3%
  11. 95% of data fits within 20 of the p 0.954 95.4%
  12. 99.7% of data fits within 30 of the p 0.997 99.7%
  13. Simple problems solved using the imperial rule - firstly, make a table out of the rule 3 to- -2 to- -1 to 2 2% 1 14% 34% Otol 34% Ito 2 14% 2 to 3 2% 0% he heights of students at a ollege were found to follow bell-shaped distribution ith p of 165cm and o of 8 at proportion o stu ents x—g first standardise 16% O -30 -20 -000 20 30 157-165 first 157cmis or la below the g
  14. Simple problems solved using the imperial rule - firstly, make a table out of the rule 3 to- -2 to- -1 to 2 2% 1 14% 34% Otol 34% Ito 2 14% O 2 to 3 2% 0 26 36 0% he heights of students at a ollege were found to follow bell-shaped distribution ith p of 165cm and o of 8 bove roughly what heigh re the tallest 2% of the 165 0 -36 -26 -o 181 cm
  15. Task — class 10 minutes finish for homework • Exercise A Page 76
  16. The Bell shape curve happens so when recording continuous random variables that an equation is used to model the shape exactly. Put it into your calculator and use the graph function. Sometimes you will see it using phi =
  17. Luckily you don't have to use the equation each time and you don't have to integrate it every time you need to work out the area under the curve — the normal distribution probability There are normal distribution tables Tables of the Normal Distribution Probability Content | 0.00 o . 01 0.02 0.03 o. 04 o. 05 0.06 0.07 o . 08 0.0 0.1 0.2 0.3 from -oo to Z z 4 | 0.5000 I o. 5398 | 0.5793 1 0.6179 o. 6554 o. 5040 o. 5438 o. 5832 o. 6217 o. 6591 o . 5080 o. 5478 o. 5871 o. 6255 o. 6628 o. 5120 o. 5517 o. 5910 o. 6293 o. 6664 0.5160 o. 5557 o. 5948 o . 6331 o. 6700 o . 5199 o. 5596 o. 5987 o. 6368 o. 6736 o. 5239 o. 5636 o. 6026 o. 6406 o. 6772 o. 5279 o. 5675 o. 6064 o. 6443 o. 6808 o . 5319 o . 5714 o. 6103 o. 6480 o. 6844 o. 09 o. 5359 o. 5753 o. 6141 o. 6517 o. 6879
  18. 6
  19. How to read the Normal distribution table (0.24) means the area under the on the left of 0.24 and is this curve value here: T les of the Normal Distribution Probabili Content from -00 t Z | 0.00 o . 01 0.02 0.03 o. 05 0.06 0.07 o . 08 o. 09 0.0 0.1 0.2 0.3 0. z 4 | 0.5000 I O. 5398 | 0.5793 1 0.6179 o. 6554 o. 5040 o. 5438 o . 5832 o. 6217 o. 6591 o . 5080 o. 5478 o. 5871 o. 6255 o. 6628 . 04 o. 5120 o. 60 o. 5517 0 57 o. 5910 o. 5948 o. 6293 o . 6331 o. 6664 o. 6700 o. 5199 o. 5596 o. 5987 o. 6368 o. 6736 o. 5239 o. 5636 o. 6026 o. 6406 o. 6772 o. 5279 o. 5675 o. 6064 0. 6443 o. 6808 o . 5319 o . 5714 o. 6103 o. 6480 o. 6844 o. 5359 o. 5753 o. 6141 o. 6517 o. 6879
  20. Values of D istntnn.ion
  21. Values of • (this is for the left) • Area = 1-0.78814 = 0.21186 -4 Disüünn.icm -3
  22. Values of O(z) 0(-?.00) -?- 0(1.00) -?-0.84134 -0.15866 Shaded area 0(1.5)- 0(-?.00) 0.93319 - 0.15866 0.77453
  23. Task • Exercise B page 79
  24. Solving Problems using the tables NO MAL DISTRIBUTION • The area under the curve is the probability of getting less than the z score. The total area is 1. • The tables qive the probability for z-scores in the distribution that is mean =0, s.d. — 1. ALWAYS SKETCH A DIAGRAM • Read the question carefully and shade the area you want to find. If the shaded area is more than half then you can read the probability directly from the table, if it is less than half, then you need to subtract it from 1. NB If your z-score is negative then you would look up the positive from the table. The rule for the shaded area is the same as above: more than half — read from the table, less than half subtract the reading from 1.
  25. You will have to standardise if the ean is not zero and the standard deviation is not one first standardise
  26. Task • Exercise C page 168
  27. Normal distribution problems in reverse • Percentage points table on page 155 • Work through examples on page 84 and do questions Exercise D on page 85
  28. Key chapter points • The probability distribution of a continuous random variable is represented by a curve. The area under the curve in a given interval gives the probability of the value lying in that interval. If a variable X follows a normal probability distribution, with mean p and standard deviation o, we write (p, 02) The variable Z= c; is called the standard normal variable corresponding to X
  29. Key chapter points cont. • If Z is a continuous random variable such that ZNN (0, 1) then • The percentage points table shows, for probability p, the value of z such that

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