The table below shows the number of students who are varsity and junior varsity athletes. Find the probability that a student is a senior given that he or she is a varsity athlete.

Class Varsity JuniorVarsity

Totals

Freshman 7 269 276

Sophomore 22 262 284

Junior 36 276 312

Senior 51 257 308

Totals 116 1064 1180

𝑃 (𝑆|𝑉 )=𝑃 (𝑉∧𝑆)𝑃 (𝑉 ) ¿

51116 ¿0.181 or 18.1%

Warm-Up 5/06

Rigor:You will learn how to find the measure of

center, spread, and position.

Relevance:You will be able to find the measure of center, spread, and position of real world data sets.

0-9 Measure of Center, Spread,

and Position

Measures of Center, Spread and Position

Statistics is the science of collecting, organizing, displaying and analyzing data in order to draw conclusions and make predictions.

Descriptive Statistics is the branch of statistics that focuses on collecting, summarizing and displaying data.

Population is the entire group from which you are collecting data.

Variable is a characteristic of the population that can assume different values called data.

Univariate Data is data that includes only 1 variable.

Sample is a smaller group of the population (subset) when it is not possible to obtain data from every member of the population.

Univariate data are often summarized using a single number to represent what is typical of the population.

Measures of average are also called Measures of Central Tendency or Measures of Center.

Key Concept: Measures of CenterThe Mean is the sum of the values divided by the total number of values. sample mean population mean

The Median is the middle value or the average of the two middle values when the data is arranged in numerical order.

The Mode is the value(s) that appear most often in the set of data.

Example 1: The number of milligrams of sodium in a 12-ounce can of ten different brands of regular soda are shown. Find the mean, median and mode.

50, 30, 25, 20, 40, 35, 35, 10, 15, 3510,15, 20, 25, 30, 35, 35, 35, 40, 50

mean

¿2 9510

=29.5 milligrams

median

¿3 0+35

2=32.5milligrams

mode 35 occurs most often, so the mode is 35 milligrams.

Since two very different data sets can have the same mean, statisticians also use measures of spread or variation to describe how widely the data values vary and how much the values differ from what is typical.

Key Concept: Measures of SpreadThe range is the difference between the greatest and the least values in a data set.

Key Concept: Measures of SpreadThe variance in a set of data is the mean of the squares of the deviations or differences from the mean. sample

variance population variance

The standard deviation in a set of data is the average amount by which each individual value deviates or differs from the mean. It is the square root of the variance. sample SD population SD

Example 2: Two classes took the same midterm exam. The scores of 5 students from each class are shown. Both sets of scores have a mean of 84.2. a. Find the range, variance and standard deviation for Class A. b. Find the range, variance and standard deviation for class Bc. Compare your results. Class A Class B

85, 76, 92, 88, 80 75, 85, 95, 98, 68

x x-

85 85 – 84.2 = 0.8 (0.8)2 = 0.64

76 76 – 84.2 = –8.2 (–8.2)2 = 67.24

92 92 – 84.2 = 7.8 (7.8)2 = 60.84

88 88 – 84.2 = 3.8 (3.8)2 = 14.44

80 80 – 84.2 = –4.2 (– 4.2)2 = 17.64

Sum = 160.8

Range 92 −76=16

Variance 𝑠2=¿¿¿¿

160.85 −1

≈ 4 0.2

Standard Deviation

s=√𝑠2¿√ 160.85 −1

≈ 6 .3

Example 2: Two classes took the same midterm exam. The scores of 5 students from each class are shown. Both sets of scores have a mean of 84.2.

b. Find the range, variance and standard deviation for class B

Class A Class B

85, 76, 92, 88, 80 75, 85, 95, 98, 68

Range 98 − 68=30

Variance (𝑆𝑥)2≈ 1 63.7

Standard Deviation Sx≈ 12. 7945

To find measure of center and spread using a graphing calculator, first clear all list. (2ND + 7 ENTER 2) Press STAT ENTER to enter each data value. Then press STAT and arrow over to the CALC menu and select 1-Var Stats ENTER .

maxX – minX

Example 2: Two classes took the same midterm exam. The scores of 5 students from each class are shown. Both sets of scores have a mean of 84.2.

c. Compare the sample standard deviations of Class A and Class B.

Class A Class B

85, 76, 92, 88, 80 75, 85, 95, 98, 68

Class A

Standard Deviation

Class B

6 .3 12. 8

There is more inconsistency in the sample scores from class B.

Quartiles are 3 position measures that divide a data set arranged in ascending order into four groups, each containing about 25% of the data.

The median marks the 2nd quartile Q2 and separates the data into upper and lower halves.

The 1st or lower quartile Q1is the median of the lower half.

The 3rd or upper quartile Q3 is the median of the upper half.

The 3 quartiles along with the minimum and maximum values are called the five-number summary of a data set.

Example 3: The number of hours that Liana , worked each week for the last 12 weeks were 21,10,18,12,15,13,20,20,19,16,18 and 14. a. Find the minimum, lower quartile, median, upper quartile and the maximum of the data set.

Use your calculator.

minimumlower quartilemedian upper quartilemaximum

minX =10Q1= 13.5med =17Q3 = 19.5maxX = 21

b. Interpret this five-number summary.

Over the last 12 weeks, liana worked a minimum of 10 hours anda maximum of 21 hour. She worked less than 13.5 hours 25% ofthe time, less than 17 hours 50% of the time, and less than 19.5 hours 75% of the time.

The Interquartile Range (IQR) is the difference between Q3 and Q1; it contains about 50% of the values.

An Outlier is an extremely high or extremely low value when compared to rest of the values in the set.

To find an outlier look for data values that are beyond the upper or lower quartiles by more than 1.5 times the IQR.

𝐼𝑄𝑅=𝑄3 −𝑄1

𝑄1 −1.5 𝐼𝑄𝑅∧𝑄3+1.5 𝐼𝑄𝑅

Example 4: The number of minutes each of 22 students in a class spent working on the same algebra assignment is shown below.15,12,25,15,27,10,16,18,30,35,22,25,65,20,18,25,15,13,25,22,15,28a. Identify any outliers in the data. Use your calculator.

b. Find mean, median, mode, range, and standard deviation of the data set both with and without the outlier. Describe the effect on each measure.

minX = 10Q1= 15med = 21Q3 = 25maxX = 65

𝐼𝑄𝑅=𝑄3 −𝑄1 𝑄1 −1.5 𝐼𝑄𝑅∧𝑄3+1.5 𝐼𝑄𝑅¿2 5−15¿10 15 −1.5 (10)25+1.5(10)

0 40Outlier for the data set is .

Data set Mean Median Mode Range Standard Deviation

with outlier 22.5 21 15 55 11.2

without outlier 20.5 20 15 25 6.5

AssignmentProb/Stats #4 WS, 1-11 All

Prob/Stats Test Friday 5/16