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Examples from Blu4, Chapter 3 Background skills – putting data into a “list” on the TI-84 and related activities, such as clearing a list, are assumed. If you need to know about them, refer to this handout: . ..\TI-84_General\TI-84_Lists.pdf Example 3-1 – Find the mean Put the data into a list. We used list L1. Press the STAT key. → to the CALC submenu. 1:1-Var Stats ENTER. The mean is the value shown at the 𝑥̅ = line. We round off the mean to one more decimal place than what is shown in the data. Our final answer is 𝑥̅ = 30.7. We say “𝑥̅ = 30.7”, not just the number “30.7” by itself. The other values on this screen will be discussed in other examples. Example 3-2 – Find the mean – using a list other than list L1 Put the data into a list. We’re going to use List L2. STAT, →, 1:1-Var Stats again, but don’t press ENTER yet! The 1-Var Stats command uses list L1 by default. We tell it to use List L2: 2ND STAT 2:L2. Then ENTER. We answer with “𝑥̅ = 5.77”, rounding to one place more than the data. Document1 4/30/2017 9:32 PM - D.R.S. Example 3-2 redone with { list } rather than a stored list 2ND STAT ← 3:mean( Put the list inside squiggly braces. 2ND ( for the starting “{“ Then the values separated by commas 2ND ) for the ending “}” Close the right parenthesis ) Aand press ENTER. Example 3-3 – The mean of a frequency distribution Put the class midpoints into one list (we used L1, the values are column “C” in the book’s byhand calculation)) and the corresponding frequencies into another list (we used L2, the values are in column “B” in the book’s by-hand calculation.) STAT → 1:1-Var Stats as usual, but next comes something different: 2ND STAT 1:L1 , the comma key 2ND STAT 2:L2 Quick typing check: the two lists must have the same number of elements; they end at the same line in the lists screen. This means “List L1 contains the values and List L2 contains the frequencies”. Press ENTER Results: 𝑥̅ = 24.5 (we would have rounded to one decimal place anyhow). Document1 4/30/2017 9:32 PM - D.R.S. Examples 3-4, 3-5, 3-6, 3-7, and 3-8 – Find the median. We put the data for all five of these problems into lists L1 through L5. The 1-Var Stats command gives We do 1-Var Stats for each of the us more information than fits on five lists: STAT → 1:1-Var Stats a screen. Observe the down arrow at the bottom of the first 2ND STAT #:L#. page of data. You must scroll down to get to the median. See on the line that says “Med=”. This is Example 34. Here are the second page results with the median for Example 35. And Example 3-7 And Example 3-8 And Example 3-6. Examples 3-9, 10, 11, 12, 13 – Find the Mode The TI-84 doesn’t compute Mode for us. Example 3-14 – Find both the Mean, Median, and Mode The 1-Var Stats results give the Mean and Median (scroll to the second screen of results for the Median), so the list only needs to be typed once, the 1-Var Stats command only needs to be issued once, and many questions are answered. The Mode must be done manually. Document1 4/30/2017 9:32 PM - D.R.S. Example 3-15, 16 – Find the Midrange Put the data into lists. We used list L1 for Example 3-15 and list L2 for Example 3-16. Do 1-Var Stats for each list. Scroll to the second screen. Here are results for Ex. 3-15 Here are 1-Var Stats for Example 3-16. The 1-Var Stats does not compute midrange for us. But we can finish by hand: Example 3-15: Example 3-16: Midrange = Midrange (1+8) = 2 = 4.5 Midrange = (1+16) 2 = 8.5 (𝑚𝑖𝑛𝑋+𝑚𝑎𝑥𝑋) 2 Example 3-17 – Weighted Average This works exactly the same as finding the mean of a frequency distribution. The “weight” of each value acts like the “frequency” did in the frequency distribution. We put the grade points (A = 4, B = 3, C = 2, D = 1, F = 0) into list L1 and the weights (the corresponding credit value of each course) in list L2. Then we do 1-Var Stats L1,L2 The weighted mean is in the 𝑥̅ . The GPA is rounded to 2.67 or 2.7. Document1 4/30/2017 9:32 PM - D.R.S. Examples 3-18, 19, 20, 21, 22 – Variance and Standard Deviation These examples all use the lifetime of two different brands of paints and make comparisons.. We use list L1 for the sample of lifetimes for Brand A and list L2 fo rthe sample of lifetimes for Brand B. Here are results for Brand A Here are results for Brand B Remarks Both Brands have mean lifetime 𝑥̅ = 35 but the underlying data are greatly different between the two brands. 𝑆𝑥=sample standard deviation (the (𝑛 + 1) denominator) and 𝜎𝑥=population standard deviation (the 𝑛 denominator). If you need either of the variances, take the standard deviation and square it. Another way to get the variance Suppose we want a super-precise value for the sample variance of Brand B and we don’t want to retype 7.071067812. With the “1-Var Stats L2“ results fresh in the calculator’s memory, do the following: VARS 5:Statistics 3:Sx X2 ENTER . The sample variance, (Sx)2 in calculator language and 𝑠 2 in proper statistics language, is 50. Document1 4/30/2017 9:32 PM - D.R.S. Example 3-23 – Sample variance and Sample standard deviation The first bunch of examples on variance and standard deviation introduced four ideas all at once: With the 𝑛 in the denominator of the formula Population variance Population standard deviation with the (𝑛 − 1) in the denominator of the formula Sample variance Sample standard deviation But usually it’s going to be the sample standard deviation that we’re interested in. With our data for Example 3-23 in List L1, STAT → 1:1-Var Stats ENTER It gives us BOTH. We read the problem carefully and see that the sample variance and sample standard deviation are requested. Sample Standard Deviation Is calculator’s Sx, properly named 𝑠 or 𝑠𝑥 . To get the variance, either retype and square the value, with more decimal places for more accuracy. Or use the VARS 5:Statistics 3:Sx and square that value. Population standard deviation is the calculator’s σx, properly called 𝜎 or 𝜎𝑥 . Final answers: The sample variance is 𝑠 2 = 1.28 and the sample standard deviation is 𝑠 = 1.13. Document1 4/30/2017 9:32 PM - D.R.S. Example 3-24 – Variance and Standard Deviation of a frequency distribution When they don’t explicitly say “find the sample standard deviation”, they mean the sample standard deviation. Don’t use the population standard deviation unless they say they want that one in particular. Recall the earlier example when we used the 1-Var Stats L1,L2 to find the mean of a frequency distribution. The same procedure does all the other statistics for a frequency distribution, too. Put the data values into list L1 (the Midpoint column in their byhand example) and the frequencies into list L2. STAT → 1:1-Var Stats 2ND STAT 1:L1 , (comma) 2ND STAT 2:L2 ENTER Since it’s implied that it’s the sample standard deviation they want, we use the Sx=8.287593772 and square it (VARS 5:Statistics 3:Sx X2 ENTER) to get 68.68421053. Our answers are: The (sample) standard deviation is 𝑠 = 8.3 and the (sample) variance is 𝑠 2 = 68.7. Examples 3-25, 26 – The coefficient of variation This is a plain arithmetic calculation; there is no built-in TI-84 variable to give this value. Examples 3-27, 28 – Chebyshev’s Theorem This involves algebra and arithmetic; there is no built-in TI-84 way to solve these problems. Examples 3-29, 30 – Standard scores (z scores) This is a plain arithmetic calculation; there is no built-in TI-84 variable to give this value. Examples 3-31, 32, 33, 34, 35 – Percentiles There is no built-in TI-84 method for doing these problems. Example 3-36 – Quartiles Put the data into a TI-84 list and use 1-Var Stats to find 𝑄1 , 𝑄2 , 𝑄3 . (You need to scroll onto the second screen of 1-Var Stats results.) Answers: 𝑄1 the first quartile, is 9. 𝑄2 is the median, which is 14. 𝑄3 is the third quartile, which is 20. Example 3-37 – Outliers There is no direct TI-84 way to find outliers. But you can use the 1-Var Stats to find 𝑄1 and 𝑄3 , and from there the Interquartile Range and the low and high values of the interval for non-outliers is easily found using arithmetic. Examples 3-38, 39 – The Five-Number Summary and Boxplots See separate document: Boxplot.pdf. Document1 4/30/2017 9:32 PM - D.R.S.