# Chapter 5: The normal approximation for data

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1 Chapter 5: The normal approximation for data Context Normal curve 3 Normal curve Areas under the normal curve Normal approximation 6 Standard units Examples % - 95% rules Normal approximation for data Percentiles and percentile ranks 11 Percentiles Quartiles Percentile rank Percentile vs. percentile rank Percentiles and the normal curve Change of scale 17 Motivation Effects of change of scale Data that don t follow the normal curve 20 How to summarize data that don t follow the normal curve?

2 Context We still look at summarizing data Many histograms follow the normal curve, if they are drawn in standard units So we ll discuss what standard units are And you ll learn to work with the normal curve Then we ll combine these two. We will use the normal curve to estimate the percentage of entries of a real data set in a given interval Other new subjects: percentiles, percentile-ranks, box-plot, interquartile range 2 / 21 Normal curve 3 / 21 Normal curve Other names: Gaussian curve, bell curve Discovered in 1720 by Abraham de Moivre See picture on overhead Important properties: the curve is symmetric about 0 the total area under the curve is 100% the curve is always above the horizontal axis 4 / 21 2

3 Areas under the normal curve Areas under the normal curve to memorize: Area between -1 and 1: 68% Area between -2 and 2: 95% Area between -3 and 3: 99% We can find other areas under the curve using: the table in the back of the book total area under the curve is 100% curve is symmetric about zero How to do this? sketch normal curve and shade area of interest build the area from areas that you know from the table See examples on overhead (Section 5.2, Exercise set B) 5 / 21 Normal approximation 6 / 21 Standard units Why use standard units? Many histograms follow the normal curve if they are drawn in standard units How to convert a value to standard units? Compute how many SDs the value is above or below average. Values above average have a plus sign, values below average have a minus sign. 7 / 21 Examples Example: women in HANES study. Average height is 63.5 inches, SD is 2.5 inches. Convert the following heights to standard units inches. Answer: 68.5 inches is 5 inches above the average. That is two SDs above the average.her height in standard units is inches. Answer: 61 inches is 2.5 inches below the average. That is one SD below the average. Her height in standard units is inches. Answer: 63.5 inches is right at the average. So this value is zero SDs above the average. Her height in standard units is 0. See Figure 2 on page 81 8 / 21 3

4 68% - 95% rules 68% rule: for many lists 68% of the entries are between average - SD and average + SD. Where does this rule come from? convert the interval to standard units: -1 to 1 the area under the normal curve between -1 and 1 is 68% if the histogram follows the normal curve, the area under the histogram is about 68% 95% rule: for many lists 95% of the entries are between average - 2 SDs and average + 2 SDs. Where does this rule come from? convert the interval to standard units: -2 to 2 the area under the normal curve between -2 and 2 is 95% if the histogram follows the normal curve, the area under the histogram is about 95% 9 / 21 Normal approximation for data We have histogram that follows that normal curve. We know the average and SD. We want to find the percentage of entries in a certain interval. Solution: Step 1: Draw the number line and shade the interval of interest. Mark the average on the line. Step 2: Convert to standard units. Draw a new line in standard units, and shade the interval of interest. Step 3: Sketch the normal curve, and find the area above the shaded interval of Step 2. See examples on overhead (Section 5.2, Exercise set C) 10 / 21 4

5 Percentiles and percentile ranks 11 / 21 Percentiles We can also summarize data by looking at percentiles. What are percentiles? 1st percentile = number such that 1% of the entries are smaller than the number, and 99% are 10th percentile = number such that 10% of the entries are smaller than the number, and 90% are 25th percentile = number such that 25% of the entries are smaller than the number, and 75% are 50th percentile = number such that 50% of the entries are smaller than the number, and 50% are 75th percentile = number such that 75% of the entries are smaller than the number, and 25% are etc. 12 / 21 Quartiles 1st quartile = number such that one quarter of the entries are smaller than that number, and three quarters are = 25th percentile 2nd quartile = number such that two quarters of the data are smaller than that number, and two quarters are = 50th percentile = median 3rd quartile = number such that three quarters of the entries are smaller than that number, and one quarter is = 75th percentile interquartile range = 3rd quartile - 1st quartile See summary of homework grades, and example with Table 1 on page / 21 Percentile rank The percentile rank of a value is the percentage of entries smaller than that value. Examples: the percentile rank of the highest homework score is 100% the percentile rank of the median homework score is 50% 14 / 21 5

6 Percentile vs. percentile rank Percentile and percentile rank are each other s opposites Example income data: The 10th percentile is \$10,200. It is the number such that 10% of the entries are smaller than that number. The percentile rank of \$10,200 is 10%. It is the percentage of entries smaller than \$10,200. Yet another way of saying this: An annual income of \$10,200 puts you at the 10th percentile of the income distribution Remember: A percentile is a number A percentile rank is a percentage 15 / 21 Percentiles and the normal curve If a histogram follows the normal curve, then the normal curve can be used to estimate percentiles Method: sketch a normal curve; find the right value of z, using the normal table z is given in standard units; convert it back to the units in the problem See examples (Section 5.5, Exercise set E) 16 / 21 Change of scale 17 / 21 Motivation Changes of scale often occur: length: cm - inches - feet temperature: Fahrenheit - Celsius Such changes of scale consist of: multiplying all entries by a constant adding a constant to all entries or both of the above What do such changes do to the average, SD and standard units? See examples on overhead 18 / 21 6

7 Effects of change of scale If we add a constant to all entries in a list, then the average increases by this constant the SD does not change the standard units do not change If we multiply all entries in a list by a positive number, then the average is multiplied by this number the SD is multiplied by this number the standard units do not change 19 / 21 Data that don t follow the normal curve 20 / 21 How to summarize data that don t follow the normal curve? Don t use the normal curve! Good ways to summarize such data are: the histogram a table with percentiles (like for the income data) the 1st quartile, median and 3rd quartile (like for the homework grades) box-plot: box given by 1st and 3rd quartile: contains the middle 50% of the data median is given as a line in the box it also gives some information on entries that fall outside the box see example for homework grades 21 / 21 7

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