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6.7 `Measures of central tendency Mean or average of a numerical set of data is where you add all the numbers together and divide by the total number. Mean is often referred to as βx-barβ and the symbol looks like π₯. Median of a numerical set of date is the middle number when the values are in order from least to greatest. If there are 2 middle numbers then you average the 2 to find the median. Mode of a numerical set of data is the value that occurs the most often. Range of a numerical set of data is the Greatest number minus the smallest number. Example: Find the mean, median, mode, and range of the following data set. 65, 68, 71, 77, 81, 82, 86, 88, 93, 93, 95, 97 Mean: 83 Median: 84 Mode: 93 Range: 32 Practice At banks county high school for physical fitness week all the ninth graders height was measured in inches and written down. The following lists are for the ninth grade boys and the ninth grade girls. Boys: 59, 60, 61, 61, 62, 63, 63, 65, 66, 66, 66, 67, 68, 70, 72 Girls: 52, 52, 54, 55, 55, 55, 55, 56, 56, 57, 59, 60, 60, 63, 64, 66 1. Find the range of both sets of data (boys and girls). Which has a larger range? 2. Find the mean and the median of the boys heights. Which is larger the mean or the median? 3. Find the mean of the girls heights. 4. Do either set of data have a mode? If so what is it? 5. Combine the girls and the boys data, then find the mean and median for the entire data set. How does the mean of all the numbers compare with the mean of just the boys, or the mean or just the girls? ο Mean Absolute Deviation (MAD) of a numerical set of data is the average deviation of the data from the mean. ο The MAD is calculated by first finding the mean (π₯) Then subtracting the mean from a data value. Once you have done this for each data value you then take the absolute value of all the new values. Lastly add these new values to each other and divide by the total number. ο Example: Find the MAD for the following set of data: 11, 5, 7, 5, 6, 10, 12 X 11 Step 1: Find π₯ 5 56 =8 7 Step 2: do x-π₯ for each value. Step 3: Find the absolute value for each number obtained in step 2. Step 4: Find the average of the numbers from step 3. 7 5 6 10 12 π₯= MAD = X - Μ x Absolute Value Quartiles A quartile is a type of median of an ordered set of data. However it is not the middle number like a median is. If you divide a set of data in half (a upper half, and a lower half) the upper quartile is simply the median or middle of the upper half of data, and the lower quartile is the median or middle of the lower half of data. Example: Looks at the data set: Lower Half Upper Half 22,23,24,25,26,27,27,27,28,30,31 Q1 Lower Quartile Median Interquartile Range (IQR) is Q3 β Q1 In the problem above the IQR is 28-24=4 Q3 Upper Quartile ο Find the mean absolute deviation for each data set below 1) 5, 9, 11, 4, 12, 15, 7 2) 12, 8,9,4,3,2,4 3) 8,9,7,11,17,15,10 4)14,6,5,15,9,11,3 ο Find the Q1, Q3, and IQR 5. 41, 37, 58, 62, 46, 33, 74, 51, 69, 81, 55 6. 182, 117, 149, 172, 161, 105, 179, 142, 187, 170, 155, 129 ο The five number summary is made up of the minimum, Q1, Median, Q3, and Maximum values. ο You can find this on your calculator by following these steps: 1st: Find the DATA button, and press it. (Note if there are any numbers typed into the columns you need to delete them before typing in your numbers. 2nd: After you have typed in your numbers press the green 2nd button and then the DATA button, followed by the ENTER button 4 times. 3rd: This will bring up a list of important values from your data set Note: #2 is the mean, #7,8,9,A,B are the 5 number summary. ο 5 Number Summary: Min, Q1, Med, Q3, Max ο Box and whiskers plot is a graph of the 5 number summary. Q1 Med Min Q3 Max ο Example: Find the 5 numbers Sum, The IQR, and make a box and whiskers plot for the data: ο 42, 42, 44, 45, 46, 46, 48, 49, 52, 53, 56 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 ο Practice: Create a box and Whiskers plot for the following problems. 1. 11, 16, 18, 17, 20, 10, 14, 10, 17, 12, 15 2. 10, 63, 52, 40, 8, 12, 73, 49, 26, 57, 32, 19, 20, 17, 55 Frequency Tables ο The most common type is called a Dot Plot. In a Dot Plot each data value is represented by a dot on a graph. ο In the dot plot above each dot represents the number of times that value occurs in the data set. For example since 5 has three dots above it that means the number occurs three times in the data set. ο Example: Create a dot plot for the following data set: 66, 60, 68, 61, 62, 68, 66, 63, 62, 65, 65, 64, 66, 62, 66, 67, 68, 64, 65 60 61 62 63 64 65 66 67 68 Histograms A histogram is a type of frequency table. However, instead of the x-axis representing a single value it represents a range of values. For instance in the histogram to the right the yellow bar represents the students who scored between a 40 β 59 on the final exam. You can see about 20 students fall into that range of scores. Note: We do not know any exact scores from these 20 students we only know that they fall between 40 β 59. The scores could be anywhere in that range. Also note: that creating the intervals is the most important part of a successful histogram. You will not always use the same interval range. Example: In the chart are scores for the 2010 Olympic figure skaters. Create a histogram using the data. 1st thing we need to do is decide what our intervals will be. Since the smallest number is 171 and the largest is 229 lets start at 170 and go up by 10s. So we have the following: 170-179 180-189 190-199 200-209 210-219 220-229 Figure Skater Scores Yu-Na Kim 228.56 Mao Asada 205.5 Joanie Rochette 202.64 Mirai Nagasu 190.15 Miki Ando 188.86 Laura Lepisto 187.97 Rahail Flatt 182.49 Akiko Suzuki 181.44 Alena Leonova 172.46 Ksenia Makarova 171.91 1.Using the dot plot to the right find the mean and the median of the data. For 2-7 use the following information. In Mr. Tysonβs math class there are 35 students. They each measured their height in inches and made a list for the whole class. 55, 55, 55, 56, 56, 56, 56, 57, 57, 58, 58, 59, 60, 60, 60, 60, 60, 61, 61, 62, 62, 63, 63, 63, 63, 63, 64, 65, 65, 65, 66, 67, 70, 72, 78 2. Create a dot plot for the heights. 3. Create a histogram for the heights. 4. Using your plots and graphs from 3,4, estimate what the mean and median for the data. 5. Find the actual mean of the data. How close was your guess in #4? 6. Create a box and whiskers plot using the 5 number summary. 7. How close was your answer in #4 to the actual median you found in #6? Finding Lines of best Fit Review writing equations of linear lines ο Remember Linear Lines mean straight lines ο The basic formula for all linear lines is y=mx+b ο m=slope and b=y-intercept Find the equation of the line in the graph. ο A line of best fit is a line that follows the pattern of the data. It goes through or close to as many of the data values as possible. ο For each line of best fit there are confidence values for how accurately the line matches the data. There are 4 main confidence values: No correlation Weak Strong Very Strong In addition to confidence values for the line of best fit on a scatter plot we also have what is called correlation values. The correlation is the direction of the slope: Practice: Tell what confidence and correlation the scatter plots have. 1. 2. 3. What is the confidence and correlation of the line? 4. Write the equation of the line. 5. Predict the grade of a student who studied for 6 hours. 70 75 80 85 2. Draw the line of best fit 60 65 1. Create a scatter plot of the data 90 95 ο Using lines of best fit to predict or project what will happen if the trend of the line continues. 1 2 3 4 5 6 2 way Frequency Tables ο A 2 way frequency table relates 2 categories together, and puts the values into a readable chart on what the categories have in common. ο Relative frequency means the to find the percent of a category compared π πππ‘πππππ¦ to one of the totals. This is found by % = π π‘ππ‘ππ Like Action Movies Do Note Like Total Action Like Comedy 450 138 588 Do Not like Comedy 190 472 662 Total 640 610 1250 1. How many people like both action and comedy? 2. How many people like just Comedy? 3. What is the relative frequency of people who like just action to the total action lovers? 4. What is the relative frequency of people who do not like either to the total number of people? Creating a 2 way frequency Table There are 150 children at summer camp and 71 signed up for swimming. There were a total of 62 children that signed up for canoeing and 28 of them also signed up for swimming. Construct a two-way table summarizing the data. Step 1: Determine the Categories. Step 2: Use the information to fill in the spaces. Step 3: Use logic and Reasoning to fill in the blanks. ο Practice: Create a 2 way frequency table using the information below, and then answer the questions at the bottom. A class was surveyed about whether they have been to Canada or Mexico. 6 people said they have been to both. 3 people said they have only been to Mexico. 11 people said they have not been to either. There was a total of 16 people who have not been to Mexico. 1) Create a 2 way frequency Table for the data 2) How many students how only been to Canada? 3) What is the relative frequency of students who have only been to Canada to the total number of students who have been to Canada? 4) What is the relative frequency of students who have been to both to the total number of students?