MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED

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1 FOM 11 T9 STANDARD DEVIATION 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) STATISTICS = the branch of mathematics used to analyze and interpret data. 2) DATA = information, often in the form of numbers, that are gathered, organized, analyzed and interpreted. 3) DATA SET = a group of numbers to be statistically analyzed. ) POPULATION = all the members of a group to be statistically analyzed. 5) POPULATION SIZE = the number of pieces of data in a data set and is represented by n. 6) SAMPLE = some of the members of a group to be statistically analyzed. 7) SAMPLE SIZE = the number of pieces of data in a data set and is represented by n. 8) RANGE = range = largest number smallest number. Range gives a rough indication of the dispersion of the numbers within a data set. 9) LINE PLOT = a number line having dots the representing each of the numbers in a data set above the corresponding number on the number line. 10) MODE = the number in a data set that appears most often. 11) MEDIAN = the number that divides the data set into two equal upper and lower halves. 12) MEAN = the average of all the numbers in a data set. Mean is represented by these symbols: x = mean for a sample, µ = mean for an entire population. 13) OUTLIER = a number that is judged to be very far from the other data in the data set. i.e. in the data set below, 35 is the outlier because it is 20 units away from its closest number. e.g. x = 7, 3, 0, 1, 2, 5, 9, 11, 15, 35 1) STANDARD DEVIATION = a number that describing the variation/dispersion (spread) of the data relative to the mean of the data. STANDARD DEVIATION I) STANDARD DEVIATION is a number describing that describes how spread out the data is relative to the data s mean A) If the standard deviation is small, the data points are close to each other and are close to the mean. If the standard deviation is large the data points are spread throughout the range of the data with some data points being close to the mean with others being far from the mean. 1) Standard deviation represented with the symbol σ and is calculated using formulae given below. σ = ( x X )2 n or σ = ( x µ ) 2 n where σ = standard deviation, Σ = the sum of, x = piece of data, X or µ = mean and n = sample or population size a) You are not required to know or use these standard deviation formulae to answer questions. They are included in this notes package for your information only. b) Compare the formulae for standard deviation to the formula used to calculate mean. NOTICE that they are similar in that they create a number that is divided by n (sample or population size). This is an indication that the standard deviation is another way to calculate an average, specifically the average distance the data points are from the mean.

2 FOM 11 T9 STANDARD DEVIATION 2 2) SAMPLE PROBLEM 1: Calculate the standard deviation for this data set: x = 2, 3,, 7 X = x n = = 16 = σ = ( x X )2 n ( = 2 ) 2 + ( 3 ) 2 + ( ) 2 + ( 7 ) 2 ( = 2) 2 + ( 1) 2 + ( 0) 2 + ( 3) 2 σ = = = 1 = 3.5! σ! 1.9 The standard deviation of the above data set is: σ! 1.9. This means that the average distance between the each number in the data set and the mean is 1.9. B) USING THE GRAPHING CALCULATOR TO DETERMINE THE SAMPLE SIZE, MEAN, MEDIAN AND STANDARD DEVIATION 1) The graphing calculator can be used to calculate the mean, standard deviation and median of a data set. 2) USE THESE STEPS TO USE THE STAT PROGRAM OF YOUR CALCULATOR TO ANALYZE A DATA SET 1) Press the STAT button. 2) Press the 1 button to enter a data set into a list. 3) Enter the data set by entering each number then pressing the ENTER button. ) When all items in the data set have been entered press the 2nd button then MODE to QUIT this list. 5) Press the STAT button. 6) Press the button so the CALC is surrounded in black. 7) Press the 1 button to select 1: 1-Var Stats the 2nd button, then NUMBER the list for you wish to analyze. i.e. press the 2nd button then 1 the data in L1, press the 2nd button then 2 the data in L2. 8) List the mean, x = #, standard deviation, σ = #, (listed as σx = = #), median (listed as Med = #). 9) List the n = sample size = number of pieces of data in the data set. 10) Calculate the range by completing this calculation: range = max X min X NOTE: The calculator does not calculate the mode, you must determine it. 3) REQUIRED PRACTICE 1: Use the STAT function (steps 1-8 listed above) of your calculator to answer question 8 for each person on page 6 & 7 from T8 notes. {Answers are on page 5 of these notes.} C) SAMPLE PROBLEM 2: Answer these questions using the STAT function (steps 1-10 listed above) of your calculator to answer parts a) through j) listed below. a) State the mean. b) State the standard deviation. c) State the sample size. d) State the median. e) Determine the range. f) How consistent is the data? g) Identify any outlier(s). h) Answer a - f without the outlier. i) Compare the mean, standard deviation, sample size, median, range and mode you calculated with and without the outlier. Describe how the outlier impacts each statistic. j) Using the standard deviation and range, comment on how consistent the data is after the removal of any outliers. 1) For the data for the masses of soft balls given below, use the STAT function (steps 1-10 listed above) of your calculator to answer parts a) through g) listed below. Masses of soft balls (g) 153.2, 15.2, 155.3, 15.8, 15.5, 15.5, 152.0

3 FOM 11 T9 STANDARD DEVIATION 3 ANSWERS: a) x = 15.1 g b) σ = 1.03 g c) n = 7 d) median = 15.5 g e) Range = 3.3 g f) The data is fairly consistent because the standard deviation ( σ = 1.03 g) and range (3.3 g) are small compared to the mean. g) outlier = g ha) x = 15.5 g hb) σ = 0.6 g hc) n = 6 hd) median = 15.5 g he) Range = 2.1 g hf) The data is fairly consistent because the standard deviation ( σ = 0.6 g) and range (2.1 g) are small compared to the mean. i) Stat With outlier Without outlier Comparison Mean: x = g g The mean has increased by 0.36 g. Stan. dev. σ = 1.03 g 0.6 g The standard deviation has decreased by 0.39 g. n = 7 6 The sample size has decreased significantly by 1. Range 3.3 g 2.1 g The range has decreased significantly by 1.2 g. Median = 15.5 g 15.5 g The median has not changed. j) The data without the outlier is much more consistent as indicated by a smaller range and a smaller standard deviation. D) REQUIRED PRACTICE 2: Answer these questions using the STAT function (steps 1-10 listed above) of your calculator. {Answers are on page of these notes.} 1) Answer parts a) through e) given above using the data each table in question 1 on page 211 of your text. Then answer part f). a) State the sample size. b) State the mean. c) State the standard deviation. d) Determine the range. e) Determine the median. f) Which city has the most consistent weather? Justify your answer. 2) Answer parts a) through f) given above using the data each person in question 3 on page 20 of your text a) State the sample size. b) State the mean. c) State the standard deviation. d) Determine the range. e) Determine the median. f) Which person s daily texting is the most consistent? Justify your answer. 3) Read the paragraph and study the table found in the section titled EXPLORE the Math on page 210 of your text. You are tasked with helping Paulo decide which brand of battery to purchase. Use the STAT function (steps 1-10 listed above) of your calculator to answer questions a) to h) below. Create a table to compare the statistics calculated for each Brand. a) Draw a line plot. b) List any outlier(s). c) State the sample size without the outlier(s). d) State the mean without the outlier(s). e) State the standard deviation without the outlier(s). f) Determine the range without the outlier(s). g) Determine the median without the outlier(s). h) Which brand of battery should Paulo purchase? Justify your answer.

4 FOM 11 T9 STANDARD DEVIATION ANSWERS TO THE REQUIRED PRACTICE Required Practice 2 from page 3 1) Text pg. 211 Langley: a) n = 12 b) x = 9. C b) σ = 5.1 C d) range = max X min X = = 1.8 C e) median = 9.2 C Windsor: a) n = 12 b) x = 9. C b) σ = 9.5 C d) range = max X min X = = 27.2 C e) median = 9.6 C f) Langley has the most consistent weather because its standard deviation ( σ = 5.1 C) and range (1.8 C) are lower than Windsor s the standard deviation ( σ = 9.5 C) is and range (27.2 C). 2) Text pg. 20 question 3 Jackson: a) n = 30 b) x = 11.6 texts/day b) σ = 7. texts/day d) range = 27 2 = 25 texts/day e) median = 9.5 texts/day Jillian: a) n = 30 b) x = 11.7 texts/day b) σ = 6.0 texts/day d) range = 2 0 = 2 texts/day e) median = 12.0 texts/day f) Jillian s daily texting is the most consistent daily texting because her standard deviation ( σ = 6.0 texts/day) and range (2 texts/day) are lower than Jackson s standard deviation ( σ = 7. texts/day) and range (25 texts/day). 3) Text pg. 210 Brand X a) Line plot for Brand X b) outlier(s) = 3.1, 3.3, 8.1, 8.2 years c) n = 26 d) x = 5.79 years b) σ = 0.91 years f) range = max X min X = = 3. years e) median = 5.75 years Brand Y a) Line Plot of Brand Y b) outlier(s) = none c) n = 30 d) x = 5.72 years e) σ = 0.6 years f) range = max X min X = = 2.5 years e) median = 5.75 years f) Paulo should purchase Brand Y because its batteries are the most consistent as indicated by it s standard deviation ( σ = 0.6 years) and range (2.5 years) are smaller than those of Brand X the standard deviation ( σ = 0.91 years) is and range (3. years).

5 FOM 11 T9 STANDARD DEVIATION 5 Required Practice 1 from page 2 PAIGE: 1. n = range = 2% 3. The line plot is below.. mode = 33% 5. median = 3% 6. mean = X = 33.9% 7. outlier(s) = none 8. standard deviation = σ = 0.83% PATRICE: 1. n = range = 5% 3. The line plot is below.. mode = 36% 5. median = 36% 6. mean = X = 36% 7. outlier(s) = none 8. standard deviation = σ = 1.50% STAR: 1. n = range = 12% 3. The line plot is below.. mode = 36%, 38% or 39% 5. median = 37% 6. mean = X = 36.3% 7. outlier(s) = 29% 8. standard deviation = σ = 3.35% ANNA: 1. n = range = 18% 3. The line plot is below.. mode = none 5. median = 38% 6. mean = X = 36.7% 7. outlier(s) = none 8. standard deviation = σ = 5.88% MORGAN: 1. n = range = 29% 3. The line plot is below.. mode = none 5. median = 37.5% 6. mean = X = 35.8% 7. outlier(s) = 19% & 21% 8. standard deviation = σ = 9.02%

6 FOM 11 T9 STANDARD DEVIATION 6 ASSIGNMENT: PRINT THIS INFORMATION ON YOUR OWN GRID PAPER LAST then FIRST Name T9 STANDARD DEVIATION Block: Show the process required to complete each problem to avoid receiving a zero grade. Neatness Counts!!! (Marks indicated in italicized brackets.) REMEMBER TO USE GRID PAPER FOR ALL ASSIGNMENTS!!! Use the STAT functions of your calculator to answer these questions for each data set given in questions 1, 2, & 3. HINT: Organize the information for questions b - g in a table for each question. a) Create a line plot. (2) b) State sample size: n = #. (0.5) c) Calculate the range. (0.5) d) List the median. (0.5) e) Determine the mode. (0.5) f) List the mean. (0.5) g) List the standard deviation. (0.5) h) Identify any outlier(s). (1) i) If the data set has an outlier(s), re-calculate the sample size, range, median, mode, mean and standard deviation without the outlier(s). (3) j) Compare the sample size, range, median, mode, mean and standard deviation you calculated with and without the outlier(s). Describe how the outlier(s) changes the sample size, range, median, mode, mean and standard deviation. (3) 1) Use the data set Media Focus Advertising from the table found on page 208 of your text to answer questions a) through j) given above. (total = 12) 2) Use the data set Computer Rescue from the table found on page 208 of your text to answer questions a) through j) given above. (total = 12) 3) Use the data set Auto Value Sales from the table found on page 208 of your text to answer questions a) through j) given above. (total = 12) ) Name the company that supplied the data set having the outlier(s) that created the greatest statistical impact. State how the outlier(s) impacted the company s payroll sample size, range, median, mode, mean and standard deviation. (2) /38