Measures of central tendency Sections 2-3 & 2-5 Measures of central tendency Measures of position

III. Section 2-3 A. Measures of Central Tendency 1. Mean – The sum of all data points divided by the number of values. a. This one is the one that we most often think of when we say “average”. 1) It’s also the one most affected by an extreme value (either high or low). 2. Median – the middle number (or mean of two middle numbers) when the data points are put into order. a. The point which has as many data values above it as there are below it. 3. Mode – The value that happens the most often (highest frequency).

B. Shapes of Distributions III. Section 2-3 B. Shapes of Distributions 1. Symmetric – Data bunched in the middle, with equal distribution on either side. 2. Uniform – Data is spread evenly across the whole spectrum. 3. Skewed Data – Named by the “tail”. a. Skewed right means most of the data values are to the left (low) end of the range. A few extreme values to the right pull the mean in that direction. b. Skewed left means that most of the data values are to the right (high) end of the range. A few extreme values to the left pull the mean in that direction. Uniform Distribution

V. Section 2-5 – Measures of Position A. Quartiles 1. Q1, Q2 and Q3 divide the data into 4 equal parts. a. Q2 is the same as the median, or the middle value. b. Q1 is the median of the data above Q2. c. Q3 is the median of the data below Q2. d. If the list of numbers is entered into Stat-Edit on the TI-84, Stat- Calc-1 will give you these values.

V. Section 2-5 – Measures of Position A. Quartiles 2. Box and Whisker Plot a. Left whisker runs from lowest data value to Q1. b. Box runs from Q1 to Q3, with a line through it at Q2. 1) The distance from Q1 to Q3 is called the interquartile range. c. Right whisker runs from Q3 to highest data value. d. To draw a box-and-whisker plot on the TI-84, follow these steps. 1) Enter the data values into L1 in STAT Edit 2) Turn on your Stat Plots (2nd Y=), and select the plot with the box- and-whisker shown 3) Set your window to match the data a) Xmin should be less than your lowest data point. b) Xmax should be more than your highest data point. 4) Press graph. The box-and-whisker plot should appear. a) Press the Trace button and you can see exactly which values make up the Min, Q1, Median, Q3, and the Max.

B. Percentiles 1. Divide the data into 100 parts. There are 99 percentiles (P1, P2, P3, …P99) a. P50 = Q2 = the median. b. P25 = Q1 c. P75 = Q3 2. A 63rd percentile score means that this person did as well as or better than 63% of the people who took that test. 3. The cumulative frequency that we did way back in section one can help us find the percentile. C. Z-Scores 1. Also called the “standard score”, it represents the number of standard deviations that a data value is away from the mean. a. 𝑧= value−mean standard deviation = 𝑥−𝜇 𝜎 2. A z-score of less than -2 or greater than 2 is considered to be unusual. a. Remember that 95% of data points should be within 2 standard deviations of the mean (if the data is symmetrically distributed). 3. A z-score of less than -3 or greater than 3 is considered to be an outlier. a. Remember that 99.7% of data points should be within 3 standard deviations of the mean (if the data is symmetrically distributed).

Practice Problem The number of tornadoes by state in a recent year is listed. Find the data set’s mean, standard deviation, minimum, first, second and third quartiles, and maximum values. Draw a box-and-whisker plot that represents the data set. If a state experienced 19 tornadoes in that year, what would its z-score be? If a state experienced 37 tornadoes in that year, what would its z-score be? 81 1 8 69 30 34 56 54 2 6 21 14 46 136 17 23 5 71 105 39 10 40 7 4 53 27 11 19 24 63

Practice Problem The number of tornadoes by state in a recent year is listed. Find the data set’s mean, standard deviation, minimum, first, second and third quartiles, and maximum values. Draw a box-and-whisker plot that represents the data set. 𝜇=25.28; 𝜎𝑥=31.610; Minimum is 0; Maximum is 136; Q1 = 2; median (Q2) = 12.5; Q3 = 39 ALL of these numbers came from STAT – Calc – 1 on the calculator. If a state experienced 19 tornadoes in that year, what would its z-score be? 𝑧= 𝑥−𝜇 𝜎 , so 𝑧= 19−25.28 31.610 ; =−.199 If a state experienced 37 tornadoes in that year, what would its z-score be? 𝑧= 𝑥−𝜇 𝜎 , so 𝑧= 37−25.28 31.610 ; =.371 81 1 8 69 30 34 56 54 2 6 21 14 46 136 17 23 5 71 105 39 10 40 7 4 53 27 11 19 24 63

Practice Problem Draw a box-and-whisker plot that represents the data set. For the box-and-whisker plot, the left whisker runs from 0 to 2 (minimum to Q1). The box runs from 2 to 39 (Q1 to Q3), with a dividing line through it at 12.5 (Q2). The right whisker runs from 39 (Q3) to 136 (maximum). NOTE: The program I used to draw the plot considered 105 and 136 to be outliers, so did not include them in the right whisker. Notice the dots at 105 and 136 instead.

Assignments: Classwork: Pages 74–76 ; #9–12, 17–39 ODDS Homework: Pages 110 – 113; #23–41 ODDS