Warm up Honors Algebra 2 3/14/19 The advertised mass of a box of cereal is 453 grams. A quality control manager measures 12 boxes and records the following masses in grams. 450, 453, 451, 456, 455, 454, 456, 452, 453, 495, 453, 456 1. Find the mean of the data. 2. Find the median of the data. 3. Which value in the data set above is most likely to be considered an outlier? 4. How does the mean of the data set change if this outlier is removed? 5. How does the median of the data set change if this outlier is removed?

Outlier An extreme value that is much less than or much greater than the other data values Have a strong effect on the mean and standard deviation Can be a result of measurement error or may represent data from the wrong population (in these cases, the value is usually removed) In order to determine if a value is an outlier, look for data values that are more than three standard deviations from the mean

For numerical data, the weighted average of all those outcomes is called the expected value for that experiment. The probability distribution for an experiment is the function that pairs each outcome with its probability

**USE WEIGHTED AVERAGE**

Finding Expected Value (CONT) 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒: 0 3 20 +1 3 20 +2 1 5 +3( 1 2 ) 1 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒:0+ 3 20 + 2 5 + 3 2 =2.05 Expected number of successful free throws is 2.05 The total will always be 1 since it is probability!!

𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒: 0 0.75 +1 0.15 +2 0.08 +3 0.02 =0.37 The expected number of accidents is 0.37

Find the mean, median, and range of the following data sets: {19, 20, 21} {0, 20, 40} Mean: 20 Median: 20 Range: 2 Mean: 20 Median: 20 Range: 40 The two data sets have the same mean and median, but the sets are very different in the way they are spread out!!

Variation A measure of variation is a value that describes the spread of the data set. Range Interquartile range Variance Standard Deviation

Measures of Variation Variance Standard Deviation 𝜎 2 Average of the squared differences from the mean. 𝜎 The square root of the variance One of the most common measures of variation

Standard Deviation Low standard deviations indicate data that are clustered near the measures of central tendency. High standard deviations indicate data that are spread out from the center

Finding Variance and Standard Deviation Find the mean of the data, 𝑥 Find the difference between the mean and each data value, and square it Find the variance, 𝜎 2 , by adding the squares of all the differences from the mean, and dividing by the number of data values. Find the standard deviation, 𝜎, by taking the square root of the variance.

Find the mean and standard deviation. This data set represents the number of people getting on and off a bus for several stops: {6,8,7,5,10,6,9,8,4} 1. Find the mean 𝑥 = 6+8+7+5+10+6+9+8+4 9 =7 𝑝𝑒𝑜𝑝𝑙𝑒

2. Find the difference between the mean and each data value, then square it.

4. Find the standard deviation 3. Find the variance Find the average of the last data row. The average of the differences squared. 𝜎 2 = 1+1+0+4+9+1+4+1+9 9 ≈3.333 4. Find the standard deviation Find the average of the last data row. Take the square root of the variance 𝜎= 3.333 ≈1.83 𝑝𝑒𝑜𝑝𝑙𝑒

Finding Stats in a calculator!! This data set represents the number of elevator stops for several rides: {0,3,1,1,0,5,1,0,3,0} Enter the data into the STAT  EDIT table (in L1) STAT  CALC  1) 1-VAR STATS

Relevant Statistics: 𝑥 :𝑀𝑒𝑎𝑛=1.4 𝜎𝑥:𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛=1.6 𝑀𝑒𝑑:𝑀𝑒𝑑𝑖𝑎𝑛=1 𝑚𝑖𝑛𝑋:𝑚𝑖𝑛𝑖𝑚𝑢𝑚=0 𝑄1:𝐹𝑖𝑟𝑠𝑡 𝑄𝑢𝑎𝑟𝑡𝑖𝑙𝑒=0 𝑄3:𝑇ℎ𝑖𝑟𝑑 𝑄𝑢𝑎𝑟𝑡𝑖𝑙𝑒=3 𝑚𝑎𝑥𝑋:𝑚𝑎𝑥𝑖𝑚𝑢𝑚=5

Box-and-Whiskers Plot in Calculator STAT  EDIT (values in L1) 2nd Y= (STAT PLOT)  1: PLOT1 Turn on Select first box-and-whiskers plot ZOOM  9: ZOOMSTAT TRACE allows you to trace each value of the box- and-whisker plot Min, Max, Q1, Q3, and Median Show on calculator

Find the following of the data set WITHOUT the use of a calculator: {4,6,10,7,7,8,9,3,4,5} Mean Median IQR Standard deviation

1. Mean 𝑥 = 4+6+10+7+7+8+9+3+4+5 10 = 63 10 =6.3 2. Median 3, 4, 4, 5, 6, 7, 7, 8, 9, 10 6+7 2 = 13 2 =6.5

3. IQR 3, 4, 4, 5, 6, 7, 7, 8, 9, 10 𝐼𝑄𝑅=𝑄3−𝑄1=8−4=4 First quartile (Q1) 4 Third quartile (Q3) 8

4. Standard Deviation 𝜎= 𝜎 2 = 4.81 =2.19 x 3 4 5 6 7 8 9 10 𝑥 −𝑥 3.3 2.3 1.3 0.3 -0.7 -1.7 -2.7 -3.7 𝑥 −𝑥 2 10.89 5.29 1.69 0.09 0.49 2.89 7.29 13.69 𝜎 2 = 10.89+5.29+5.29+1.69+0.09+0.49+0.49+2.89+7.29+13.69 10 =4.81 𝜎= 𝜎 2 = 4.81 =2.19

Find the following of the data set WITH the use of a calculator: The temperatures for this week are represented by the data set: {55, 65, 67, 51, 47, 50, 50, 52} Mean IQR Standard deviation Variance Median

Stats: Mean: 𝑥 =54.6 IQR: 𝑄3−𝑄1:60−50=10 Standard deviation: 𝜎=6.91 Variance: 𝜎 2 = 6.91 2 =47.75 Median: 51.5 Extra practice if time