1 The Standard Deviation as a Ruler  A student got a 67/75 on the first exam and a 64/75 on the second exam. She was disappointed that she did not score as well on the second exam.  To her surprise, the professor said she actually did better on the second exam, relative to the rest of the class.

2 The Standard Deviation as a Ruler  How can this be?  Both exams exhibit variation in the scores.  However, that variation may be different from one exam to the next.  The standard deviation provides a ruler for comparing the two exam scores.

3 Summarizing Exam Scores  Exam 1 – Score: 67 – Mean: – Standard Deviation:  Exam 2 – Score: 64 – Mean: – Standard Deviation:

4 Standardizing Look at the number of standard deviations the score is from the mean.

5 Standardized Exam Scores  Exam 1 – Score: 67  Exam 2 – Score: 64

6 Standardized Exam Scores  On exam 1, the 67 was 0.87 standard deviations better than the mean.  On exam 2, the 64 was 1.17 standard deviations better than the mean.

7 Standardizing  Shifts the distribution by subtracting off the mean.  Rescales the distribution by dividing by the standard deviation.

8 Distribution of Low Temps Low Temperature ( o F)

9 Shifting the Distribution Low Temperature – 32 ( o F)

10 Shifting  Temperature ( o F) – Median: 24.0 o F – Mean: 24.4 o F – IQR: 16.0 o F – Std Dev: o F  Temp – 32 ( o F) – Median: –8 o F – Mean: –7.6 o F – IQR: 16.0 o F – Std Dev: o F

11 Shifting  When adding (or subtracting) a constant: – Measures of position and center increase (or decrease) by that constant. – Measures of spread do not change.

12 Rescaling Low Temperature ( o C)

13 Rescaling  Temp – 32 ( o F) – Median: –8 o F – Mean: –7.6 o F – IQR: 16.0 o F – Std Dev: o F  Temperature ( o C) – Median: –4.4 o F – Mean: –4.2 o F – IQR: 8.9 o F – Std Dev: 6.24 o F

14 Rescaling  When multiplying (or dividing) by a constant: – All measures of position, center and spread are multiplied (or divided) by that constant.

15 Standardizing  Standardizing does not change the shape of the distribution.  Standardizing changes the center by making the mean 0.  Standardizing changes the spread by making the standard deviation 1.

16 Normal Models  Our conceptualization of what the distribution of an entire population of values would look like.  Characterized by population parameters: μ and σ.

17

18 Describe the sample  Shape is symmetric and mounded in the middle.  Centered at 60 inches.  Spread between 45 and 75 inches.  30% of the sample is between 60 and 65 inches.

19 Normal Models  Our conceptualization of what the distribution of an entire population of values would look like.  Characterized by a bell shaped curve with population parameters – Population mean = μ – Population standard deviation = σ.

20 Sample Data

21 Normal Model

22 Population – all items of interest. Example: All children age 5 to 19. Variable: Height Sample – a few items from the population. Example: 550 children. Normal Model

23 Normal Model  Height  Center: – Population mean, μ = 60 in.  Spread: – Population standard deviation, σ = 6 in.

Rule  For Normal Models – 68% of the values fall within 1 standard deviation of the mean. – 95% of the values fall within 2 standard deviations of the mean. – 99.7% of the values fall within 3 standard deviations of the mean.

25 Normal Model - Height  68% of the values fall between 60 – 6 = 54 and = 66.  95% of the values fall between 60 – 12 = 48 and = 72.  99.7% of the values fall between 60 – 18 = 42 and = 78.

26 From Heights to Percentages  What percentage of heights fall above 70 inches?  Draw a picture.  How far away from the mean is 70 in terms of number of standard deviations?

27 Normal Model Shaded area?

28 Standardizing

29 Standard Normal Model  Table Z: Areas under the standard Normal curve in the back of your text.  On line:

30 From Percentages to Heights  What height corresponds to the 75 th percentile?  Draw a picture.  The 75 th percentile is how many standard deviations away from the mean?

31 Normal Model 25% 50% 25%

32 Standard Normal Model  Table Z: Areas under the standard Normal curve in the back of your text.  On line:

33 Reverse Standardizing

34 Do Data Come from a Normal Model?  The histogram should be mounded in the middle and symmetric.  The data plotted on a normal probability (quantile) plot should follow a diagonal line. – The normal quantile plot is an option in JMP: Analyze – Distribution.

35 Do Data Come from a Normal Model?  Octane ratings – 40 gallons of gasoline taken from randomly selected gas stations.  Amplifier gain – the amount (decibels) an amplifier increases the signal.  Height – 550 children age 5 to 19.

36 Octane Rating

37 Amplifier Gain (dB)

38

39 Nearly normal?  Is the histogram basically symmetric and mounded in the middle?  Do the points on the Normal Quantile plot fall close to the red diagonal (Normal model) line?