The Normal Distribution. MDM4U Unit 6 Lesson 2

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1 The Normal Distribution MDM4U Unit 6 Lesson 2

2 Normal Distributions Many data sets display similar characteristics The normal distribution is a way of describing a certain kind of "ideal" data set Although no real-world data is perfect, a surprising amount of natural phenomena are approimately "normal"

3 Properties of the "Bell Curve" Symmetrical no skew mean, median, mode all equal Mound / Bell Shaped peaks in the middle, slopes down towards the sides data Density Histogram s Density = normaldensity (, 30, sd )

4 Why is Normal good? The Normal Distribution is so well behaved that we can draw a curve that almost matches it This makes it very easy to measure how tall the histogram bars are The height of the bars are given by the curve that matches it This allows us to find almost eactly how much data is in each part of the distribution

5 Where are we on the curve? the mean 2σ σ +σ + 2σ

6 Where are we on the curve? one standard deviation below the mean one standard deviation above 2σ σ +σ + 2σ

7 Where are we on the curve? one standard deviation below the mean one standard deviation above 2 standard deviations below 2σ σ +σ 2 standard deviations above + 2σ

8 Area under the curve 95% 68% 34% 34% 2.25% 13.5% 13.5% 2.25% 3σ 2σ σ +σ + 2σ + 3σ

9 More Properties Approimately 68% of the data is within one standard deviation of the mean Approimately 95% of the data is within two standard deviations of the mean Approimately 99.7% of the data is within three standard deviations of the mean

10 Notation X ~ N(,σ 2 ) our data (call it X) If we want to say "this data is approimated by the standard distribution"... We should also state what the mean and standard deviation are X ~ N(,σ "is approimated by" the normal distribution 2 ) and this standard deviation or variance with this mean

11 Notation Eample The data is normal, and has a mean of 3 and a standard deviation of 2 X ~ N(3,2 2 ) The data is normal, has a mean of 5.4, and a standard deviation of 3 X ~ N(5.4,9) be careful - if there is no square, then the second number is the variance, and you need to take the square root to get the standard deviation...

12 Problem Eample Julie is an engineer who is designing roller coasters. Her roller coaster must have mass restrictions that are suitable for 95% of the population. The average adult in North America has a mass of 71.8kg with a standard deviation of 13.6kg. What range of mass should her ride accommodate?

13 Problem Eample Julie is an engineer who is designing roller coasters. Her roller coaster must have mass restrictions that are suitable for 95% of the population. The average adult in North America has a mass of 71.8kg with a standard deviation of 13.6kg. What range of mass should her ride accommodate? 1. Assume that the masses are normally distributed % of the data will fall within two standard deviations Consequently, the range will be between (13.6) = 44.6kg and (13.6) = 99 kg

14 Problem Eample All That Glitters, a sparkly cosmetic powder, is machine-packaged in a process that puts approimately 50 g of powder in each package. The actual masses have a normal distribution with: X ~ N(50.5,0.6 2 ) The manufacturers want to ensure that each package contains at least 49.5 g of powder. What percent of packages do not contain this much powder?

15 Problem Eample The manufacturers want to ensure that each package contains at least 49.5 g of powder. What percent of packages do not contain this much powder? This answer falls between 1 and 2 standard deviations below the mean.

16 The Standard Normal Curve For the standard normal curve the mean is equal to 0 = 0 the standard deviation is equal to 1 σ =1 f ( ) = 1 2πσ 2 e ( ) 2 2 / 2σ sub in f ( ) = 1 2π e 2 / 2

17 Calculating z-scores number of standard deviations is away from the mean The data z = σ The mean The deviation The z-score is the deviation divided by the standard deviation The standard deviation

18 Calculating z-scores This means 49.5 g is a mass that falls eactly 1.67 standard deviations below the mean.

19 Standard Normal Curve Tables (pg. 606 and 607) This table has negative values This table has positive values

Finding P values in the Standard Normal Table = -1.

20 Finding P values in the Standard Normal Table = P( -1.67) = Locate the number and its first decimal place in the left column 2. Go across to get more precision 3. The number in the cell is the area up to

21 = P( -1.67) = Finding P values in the Standard Normal Table This means that 4.75% of the packages of powder will have a mass of 49.5 g or below.

22 Assigned Work pg. 430 # 1 3, 7, 9

Section 3.4 Normal Distribution MDM4U Jensen

Section 3.4 Normal Distribution MDM4U Jensen Part 1: Dice Rolling Activity a) Roll two 6- sided number cubes 18 times. Record a tally mark next to the appropriate number after each roll. After rolling