Notes: 7.3 Determining Normality



Learning Target Determining Normality?I can determine if data is normal based on its distribution?To determine whether data is normal, check if it follows the Empirical Rule… Check to see 68% of the data falls within 1 standard deviation of the meanCheck to see 95% of the data falls within 2 standard deviations of the mean Check to see 99.7% of the data falls within 3 standard deviations of the mean Data that follows this pattern is normal (or approximately normal if the pattern is close to these percentages).Step 1: Plot the data (histogram or dot plot will show you the best density curve)Step 2: Calculate the mean and standard deviation of the data setStep 3: Calculate the % of values that fall within 1 standard deviation of the mean % = number of values within 1 st devtotal numver of valuesStep 4: Calculate the % of values that fall within 2 standard deviations of the mean (similar to calculation above)Step 5: Calculate the % of values that fall within 3 standard deviations of the mean (similar to calculation above)Step 6: Determine if the percent of data follows the pattern of the Empirical RuleExample 1: Determine if the following data set follows a normal distribution.The heights (to the nearest inch) of 39 students in a class are:505151525252535354545455555555555656565656575757575758585858595960606162626364 475297513208000The histogram of the data looks bell shaped and approximately symmetric:Mean = 56.4Standard Deviation = 3.4Mean + 1 st. dev = 56.4 + 3.4 = 53 to 59.8Mean + 2 st. dev = 56.4 + (2)3.4 = 49.6 to 63.2Mean + 3 st. dev = 56.4 + (3)3.4 = 46.2 to 66.6There are 26 out of the 39 values within 1 standard deviation (between 53 and 59.8). This equals 66.6% (which is close to 68%)There are 38 out of the 39 values within 2 standard deviations (between 49.6 and 63.2). This equals 97.4% (which is almost 95%)There are 39 values out of the 39 values within 3 standard deviations (between 46.2 and 66.6)This equals 100% (which is almost 99.7%)Based on the % of data within 1, 2, and 3 standard deviations, the data is approximately normal.This is time consuming!!! Luckily there is a faster way – using the calculator to produce a normal quantile plot ?I can determine if data is normal based on a normal quantile plot?Normal Quantile Plots (also called a normal probability plot) change data to z scores. If the normal quantile plot appears linear, then the data is normal.To create a normal quantile plot on the calculator:Step 1: Enter data in a list in your calculator (usually L1)Step 2: Go to StatPlot (2nd Y=) and choose Plot 1Step 3: Turn Plot1 On, select the last graph (6th choice), Data List (L1 or whatever list your data is located)Step 4: Press Zoom, then Zoom Stat (Zoom 9)Your graph should appear 203073020193000Example 2: Use the data from Example 1 (heights of 39 students). Put the data in a list and create the graph following the steps above.This is should be what you see:Since the normal quantile plot shows a linear pattern, the data is normal (approximately).Interpreting the plot:If the normal quantile plot is not approximately linear, then the data is not approximately normal.5067305715000With skewed left data (towards smaller values) the normal quantile plot opens down, below the lineWith skewed right data (towards larger values) the normal quantile plot opens up, above the line ................
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