5-Minute Check on Activity 7-85-Minute Check on Activity 7-85-Minute Check on Activity 7-85-Minute Check on Activity 7-8

Click the mouse button or press the Space Bar to display the answers.Click the mouse button or press the Space Bar to display the answers.

1. What type of an experiment is it when neither the patient nor the doctor knows what type of pill is being given?

2. List the three major components of any experimental design

3. A “sugar pill” is also known as a ________________.

4. What is the only thing that can establish cause and effect?

5. What do we call a group in the experiment which treatments are measured against?

Double-blind experiment

Randomization, replication, and control

Placebo

Well designed experiment

Control group

Activity 7 - 9

A Switch Decision

Submarine interior (unspecified class) at the Royal Naval Museum, Copenhagen, Denmark

Objectives

• Measure the variability of a frequency distribution

Vocabulary

• Standard Deviation – measures how much the data deviates from the mean

• Boxplot – statistical graph that helps visualize the variability of a distribution

• Five-number Summary – the min, quartile 1, 2 and 3 and the max of the data set

ActivityThe following sets of data are the result of testing two different switches that can be used in the life-support system on a submarine. Two hundred of each type of switch were placed under continuous stress until they failed, the recorded in hours. Switch A and B have approximately the same means and medians, as displayed by the following histograms.

Activity cont

1. What does the means and medians being the same tell us about the distributions?

Distributions are symmetric

Activity cont

2. Which distribution is most spread out?

Switch B is more spread out

Activity cont

3. Which distribution is packed more closely together around its center?)

Switch A is tighter

Activity cont

4. Which of these two switches would you choose and why? Switch A because is varies less

Activity cont

5. Determine the range of the two switches:

Switch A: 87.52 – 76.68 = 10.84Switch B: 94.03 – 65.87 = 28.16

Activity cont

6. Determine the IQR (interquartile range), which is Q3 – Q1 for each switch

Switch A: 83.47 – 80.53 = 2.94Switch B: 85.45 – 78.99 = 6.46

Activity cont

7. Write down sx for each switch (this is something we will call the standard deviation)

Switch A: 2.03 Switch B: 5.00

Measures of Spread

Variability is the key to Statistics. Without variability, there would be no need for the subject.

When describing data, never rely on center alone.

Measures of Spread:Range - {rarely used ... why?}

Quartiles - InterQuartile Range {IQR=Q3-Q1}

Variance and Standard Deviation {var and sx}

Like Measures of Center, you must choose the most appropriate measure of spread.

Standard Deviation

Another common measure of spread is the Standard Deviation: a measure of the “average” deviation of all observations from the mean.

To calculate Standard Deviation:Calculate the mean.Determine each observation’s deviation (x - xbar).“Average” the squared-deviations by dividing the total squared deviation by (n-1).This quantity is the Variance.Square root the result to determine the Standard Deviation.

Standard Deviation Properties

s measures spread about the mean and should be used only when the mean is used as the measure of center

s = 0 only when there is no spread/variability. This happens only when all observations have the same value. Otherwise, s > 0. As the observations become more spread out about their mean, s gets larger

s, like the mean x-bar, is not resistant. A few outliers can make s very large

Standard Deviation

Variance:

Standard Deviation:

Example 1.16 (p.85 of YMS): Metabolic Rates

var (x1 x )2 (x2 x )2 ... (xn x )2

n 1

sx (xi x )2n 1

1792 1666 1362 1614 1460 1867 1439

Standard Deviation

1792 1666 1362 1614 1460 1867 1439

x (x - x) (x - x)2

1792 192 36864

1666 66 4356

1362 -238 56644

1614 14 196

1460 -140 19600

1867 267 71289

1439 -161 25921

Totals: 0 214870

Metabolic Rates: mean=1600

Total Squared Deviation

214870

Variancevar=214870/6var=35811.66

Standard Deviation

s=√35811.66s=189.24 cal

What does this value, s, mean?

Example 1

Which of the following measures of spread are resistant?

1. Range

2. Variance

3. Standard Deviation

Not Resistant

Not Resistant

Not Resistant

Standard Deviation Using the TI-83

• Enter the test data into List, L1– STAT, EDIT enter data into L1

• Calculate Standard Deviation– Hit STAT go over to CALC

and select 1-Var Stats and hit 2nd 1 (L1)

– Read sx to get standard deviation

– Square sx to get variance

x is population standard deviation (and won’t be used by AFDA)

• Don’t worry about the formula we just went over

Example 2Given the following set of data:

What is the range?

What is the standard deviation?

What is the variance?

32 - 19 = 13

(3.751)2 = 14.070

3.751

19 22 23 23 23 26 26 27 28 29 30 31 32

QuartilesQuartiles Q1 and Q3 represent the 25th and 75th percentiles.

To find them, order data from min to max.Determine the median - average if necessary.The first quartile is the middle of the ‘bottom half’.The third quartile is the middle of the ‘top half’.

19 22 23 23 23 26 26 27 28 29 30 31 32

45 68 74 75 76 82 82 91 93 98

med Q3=29.5Q1=23

med=79Q1 Q3

5-Number Summary, Boxplots

The 5 Number Summary provides a reasonably complete description of the center and spread of distribution

We can visualize the 5 Number Summary with a boxplot.

MIN Q1 MED Q3 MAX

min=45 Q1=74 med=79 Q3=91 max=98

45 50 55 60 65 70 75 80 85 90 95 100

Quiz ScoresOutlier?Outlier?

Box Plots Using the TI-83

• Enter the test data into List, L1– STAT, EDIT enter data into L1

• Calculate 5 Number Summary– Hit STAT go over to CALC

and select 1-Var Stats and hit 2nd 1 (L1)

• Use 2nd Y= (STAT PLOT) to graph the box plot– Turn plot1 ON– Select BOX PLOT (4th option, first in second row)– Xlist: L1– Freq: 1– Hit ZOOM 9:ZoomStat to graph the box plot

• Copy graph with appropriate labels and titles

Determining Outliers

InterQuartile Range “IQR”: Distance between Q1 and Q3. Resistant measure of spread...only measures middle 50% of data.

IQR = Q3 - Q1 {width of the “box” in a boxplot}

1.5 IQR Rule: If an observation falls more than 1.5 IQRs above Q3 or below Q1, it is an outlier.

“1.5 • IQR Rule”“1.5 • IQR Rule”

Why 1.5? According to John Tukey, 1 IQR seemed like too little and 2 IQRs Why 1.5? According to John Tukey, 1 IQR seemed like too little and 2 IQRs seemed like too much...seemed like too much...

Outliers: 1.5 • IQR Rule

To determine outliers:

1. Find 5 Number Summary

2. Determine IQR (Q3 – Q1)

3. Multiply 1.5xIQR

4. Set up “fences”

A. Lower Fence: Q1-(1.5∙IQR)

B. Upper Fence: Q3+(1.5∙IQR)

5. Observations “outside” the fences are outliers.

Outlier Example

0 10 20 30 40 50 60 70 80 90 100Spending ($)

IQR=45.72-19.06IQR=26.66IQR=45.72-19.06IQR=26.66

1.5IQR=1.5(26.66)1.5IQR=39.991.5IQR=1.5(26.66)1.5IQR=39.99

outliers}

fence: 45.72+39.99= 85.71

fence: 19.06-39.99= -20.93

{

Example 3Consumer Reports did a study of ice cream bars (sigh, only vanilla flavored) in their August 1989 issue. Twenty-seven bars having a taste-test rating of at least “fair” were listed, and calories per bar was included. Calories vary quite a bit partly because bars are not of uniform size. Just how many calories should an ice cream bar contain?

 

Construct a boxplot for the data above.

342 377 319 353 295 234 294 286

377 182 310 439 111 201 182 197

209 147 190 151 131 151

Example 3 - Answer

Q1 = 182 Q2 = 221.5 Q3 = 319

Min = 111 Max = 439 Range = 328

IQR = 137 UF = 524.5 LF = -23.5

Calories

100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500

Example 4

The weights of 20 randomly selected juniors at MSHS are recorded below:

 

a) Construct a boxplot of the data

b) Determine if there are any outliers

c) Comment on the distribution

121 126 130 132 143 137 141 144 148 205

125 128 131 133 135 139 141 147 153 213

Example 4 - Answer

Q1 = 130.5 Q2 = 138 Q3 = 145.5

Min = 121 Max = 213 Range = 92

IQR = 15 UF = 168 LF = 108

Mean = 143.6

StDev = 23.91

Weight (lbs)

100 110 120 130 140 150 160 170 180 190 200 210 220

**

Extreme Outliers( > 3 IQR from Q3)

Shape: somewhat symmetric Outliers: 2 extreme outliersCenter: Median = 138 Spread: IQR = 15

Summary and Homework

• Summary– Variability of a frequency distribution refers to how

spread out the data is, away from center– Range is the max – the min of the data– Deviation of a data value is how far away from the

mean it is– Standard deviation is a measure of how spread out

all of the data is– Boxplot is a graph of the 5-number summary

• Homework– pg 861 – 863; problems 1, 4, 6, 7