pencil, red pen, highlighter, GP notebook, graphing calculator 4/22/13 Have out: Bellwork: 1. Solve for x. 2. Determine the following. +1 a) b) +1 CHECK!!! +1 +1 total: +1 +1

Bellwork: total: 2. Determine the following. end a) b) +1 start +1 = 21–1 + 22–1 + 23–1 + 24–1 + 25–1 = 5(2) + 5(3) + 5(4) = 20 + 21 + 22 + 23 + 24 = 10 + 15 + 20 = 1 + 2 + 4 + 8 + 16 = 45 +2 = 31 +2 total:

Introduction to Standard Deviation Range measures the spread of a set of data from low to high. For example, yesterday we discussed the range of heights of an 8th grade class from 60 to 71 inches. The spread is from 60 to 71. There are two other measures of spread. Today, we will focus on the first measure: ____________ ____________. standard deviation Standard Deviation measures the spread of the data from the mean. The larger the standard deviation, the more spread out the data is from the mean. Example #1: Suppose you are given the following data set: 4, 6, 9, 6, 5. Sketch a histogram of the data.

Frequency 4 3 2 1 data 4 5 6 7 8 9

There are 6 step to compute standard deviation:  Step 1: Determine the mean. We will use the variable (pronounced “x bar”) for the sample mean. Frequency 4 5 6 7 1 2 3 8 9 n = ____ (the number of points) 5 4 6 9 6 5 30 = ____ 6 5 5 mean Remember: standard deviation is the measure of spread from the mean. data

Step 2: Determine the distance each point is from the mean. We will use the variable x1 to represent each data. Here we have x1, x2, x3, x4, and x5. Step 3: Square the differences (–2)2 4 – 6 = 4 = –2 6 – 6 = 0 (0)2 = 0 9 – 6 = 3 (3)2 = 9 6 – 6 = 0 (0)2 = 0 5 – 6 = –1 (–1)2 = 1 Sum = 0 (always)

 Step 4: Take the sum the squares of the differences. 9 1 14 = ___ + ___ + ___ + ___ + ___ = _______  Step 5: Divide the sum of the squares by n. = ___ = ___ = ___ sum 14 2.8 5 5

Step 6: is called ___________ (the mean of the squares of the differences). Take the square root of the ____________. The answer is called the standard deviation. variance variance sigma = ______ = ___ 1.67 You may be asking yourself: “Okay, the answer is 1.67. What does that mean?” Be patient… 1.67 will make more sense by tomorrow.

Example #2: Find the standard deviation of the data set 4, 8, 2, 2.  Step 1: Determine the mean. n = ____ (the number of points) 4 = 4

Step 2: Determine the distance each point is from the mean. Step 3: Square the differences (0)2 4 – 4 = 0 = 0 8 – 4 = 4 (4)2 = 16 2 – 4 = –2 (–2)2 = 4 2 – 4 = –2 (–2)2 = 4 Sum = 0 (always)

 Step 4: Sum the squares of the differences. 16 = ___ + ___ + ___ + ___ = _______ 4 4 24  Step 5: Divide the sum of the squares by n. = ___ = ___ = ___ sum 24 6 4 4 Step 6: Take the square root of the variance. = ______ = ___ 2.45

In the blank space on your worksheet, copy this formula. Make it very large!!! We are going to identify all six steps in the formula for standard deviation. You must memorize the formula! 4 3 2 1 6 5 Complete the practice.

Practice: Find the standard deviation for the data {10, 15, 14, 15, 11}. Clearly show all steps. 4 3 = 13 2 1 (10 – 13)2 = (–3)2 = 9 (15 – 13)2 = (2)2 = 4 6 5 (14 – 13)2 = (1)2 = 1 (15 – 13)2 = (2)2 = 4 (11 – 13)2 = (–2)2 = 4 = 4.4

Now we’ll play with graphing calculators. I’ll exit this PowerPoint so I can do it with you!

***We will look at Q1 and Q3 later.*** 1–Var Stats =  sample mean Sx  sample standard deviation  standard deviation of the population n  number of data points minX Q1 Med  median Q3 maxX ***We will look at Q1 and Q3 later.*** 4 16  sum of x’s (all values) 88  sum of x’s (all values squared) 2.83 2.45 4 2 2 3 6 8

The Normal Distribution One of the most important and common distribution in statistics is the ________ ___________. It is described as _________, ____________, and __________. normal distribution symmetric single-peaked bell-shaped concave concave Notice the curve is _______ down in the middle and ________ up on the ends. The place where the curve changes concavity is called the ________ point. (Recall: y = ___ also has an ________ point.) Put a dot  on the 2 inflection points on the normal curve. The center of the normal curve is the _____, X. inflection x3 inflection mean Mark X on the curve. Normal curve animation

Same distance The inflection points are one __________ ________ on either side of the mean. Mark these two points as . standard deviation and Also mark and on the curve

Application Example # 1: The Stanford–Binet IQ Test is normally distributed with a mean of 100 and standard deviation of 10. Label , , , and 70 80 90 100 110 120 130 Determine the number of standard deviations each IQ score is from the mean: a) Bob has an IQ of 120. b) Paris has an IQ of 90. c) Sam has an IQ of 100. 2 standard deviations above 1 standard deviation below 0 standard deviations above or below

2 standard deviations above 1 standard deviation above Example # 2: Mr. Thibodeaux gave a test in his Algebra 1 class. The scores were normally distributed with a mean of 73 and a standard deviation of 8. Label , , , and 49 57 65 73 81 89 97 Determine the number of standard deviations each score is from the mean: a) Brittany scored an 89. b) Andrew scored an 81. c) John scored a 49. 2 standard deviations above 1 standard deviation above 3 standard deviations below

1 standard deviation above 2 standard deviations below Example # 3: A recent study showed that the systolic blood pressure of high school students ages 14 – 17 is normally distributed with a mean of 120 and a standard deviation of 12. Label , , , and 84 96 108 120 132 144 156 Determine the number of standard deviations each blood pressure reading is from the mean: a) Juan’s reading is 120. b) Anna’s reading is 132. c) Patricia’s reading is 96. 1 standard deviation above 2 standard deviations below 0 standard deviations above or below

Assignment #3 Find the standard deviation for the data {10, 15, 14, 15, 11} DS 39, 40ac, 41, 44, 67, 68 c) CST #57 – 64