Lesson 11.1 Normal Distributions (Day 1) Learning Goal: (S-ID.A.4) I can use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Essential Question: How does the mean and standard deviation of a data set help to estimate population percentages? Create a line-plot for each of the following data. Find the mean, median, mode, and range of each data set. Group Warm-Up Set 1: 5, 3, 4, 2, 3, 4, 1, 2, 3 Set 2: 4, 5, 1, 4, 3, 4, 5, 2, 3, 4, 5 Set 3: 1, 5, 2, 3, 1, 2, 3, 1, 2, 4, 2

Seeing the Shape of a Distribution “Raw” data values are simply presented in an unorganized list. Organizing the data values by using the frequency with which they occur results in a distribution of the data. A distribution may be presented as a frequency table or as a data display. Data displays for numerical data, such as line plots, histograms, and box plots, involve a number line, while data displays for categorical data, such as bar graphs and circle graphs, do not. Data displays reveal the shape of a distribution. Normal Distribution Symmetric Skewed Left "tail" on left Skewed Right "tail" on right

Center of Distribution Shape is one way of characterizing a data distribution. Another way is by identifying the distribution’s center. Measures of Center The mean of n data values is the sum of the data values divided by n. If x1 , x2 , ..., xn are data values, then the mean is given by: μ = x1 + x2 + .... + xn n μ - Greek letter "mu" The median of n data values written in ascending order is the middle value if n is odd, and the mean of the two middle values if n is even. Where is the mean, median and mode of a skewed distribution? Skewed Right Skewed Left

= Measures of Spread Spread of a Distribution The third way of characterizing a data distribution is by looking at the spread. Measures of Spread The standard deviation of n data values is the square root of the mean of the squared deviations from the distribution’s mean (variance). The Greek letter sigma ( σ ) represents standard deviation. Example: Measurement of Dogs. The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm. 600 + 470 + 170 + 430 + 300 = 1970 = 394 5 Distribution's Mean = Now we calculate each dog's difference from the mean. To calculate the Variance, take each difference, square it, and then average the result: The standard deviation is the square root of the variance. σ = √21,704 = 147.32... = 147 (to the nearest mm) So, using the Standard Deviation we have a "standard" way of knowing what is normal, and what is extra large or extra small.

The interquartile range, or IQR, of data values written in ascending order is the difference between the median of the upper half of the data, called the third quartile or Q3 , and the median of the lower half of the data, called the first quartile or Q1 . IQR = Q3 - Q1

Using Your Calculator 5, 3, 4, 2, 3, 4, 1, 2, 3 Set 1: Set 2: μ = σ = median = IQR = Set 2: 4, 5, 1, 4, 3, 4, 5, 2, 3, 4, 5 μ = σ = median = IQR = Set 3: 1, 5, 2, 3, 1, 2, 3, 1, 2, 4, 2 μ = σ = median = IQR =

Use a graphing calculator to compute the measures of center and the measures of spread for the distribution of baby weights and the distribution of mothers’ ages. Begin by entering the two sets of data into two lists on a graphing calculator. Birth Weight μ = σ = median = IQR = Mother's Age μ = σ = median = IQR =

Practice to Strengthen Understanding Exit Question: What does the standard deviation tell us about a data distribution? Practice to Strengthen Understanding Hmwk TenMarks and Quiz 10.1-10.2 Corrections