Describing Distributions Describing Distributions With NumbersWith Numbers

Section 1.3 Section 1.3

(mean, median, range, quartiles, IQR)(mean, median, range, quartiles, IQR)

Target Goal: I can analyze data using Target Goal: I can analyze data using shape, center and spread.shape, center and spread.

Hw:Hw: pg 70: 79, 80, 81, 84, 87, 89 pg 70: 79, 80, 81, 84, 87, 89

Measuring CenterMeasuring Center

Mean ( Mean ( : x bar: x bar)): :

The most common measure of The most common measure of center.center.

x

ixx

n

Median (M)Median (M)

The The midpointmidpoint of a distribution, the # of a distribution, the # such that half the observations are such that half the observations are smaller and the other half are larger.smaller and the other half are larger.

Resistant Measure (of center)Resistant Measure (of center)

ResistsResists influence of extreme observations. influence of extreme observations. • Mean Mean (average)(average) is is not resistantnot resistant• Median Median (midpoint) (midpoint) is resistantis resistant• The mean and median would be exactly The mean and median would be exactly

the same if the same if the distribution is exactly the distribution is exactly symmetric.symmetric.

• In a skewed distribution, the In a skewed distribution, the mean is mean is farther out in the long tail then the farther out in the long tail then the median.median.

Ex: Barry BondsEx: Barry BondsFind the median of home runs hit in Find the median of home runs hit in first 16 seasons.first 16 seasons.

• n= n=

• M= M=

16163434

How do outliers affect the How do outliers affect the median?median?

Are there any outliers? If so remove Are there any outliers? If so remove and find M.and find M.

• n= 15n= 15

• MMnewnew= 34 = 34 vs Mvs Mold old = 34= 34

Median Median is resistantis resistant

How do outliers affect the mean?How do outliers affect the mean?16, 19, 24, 25, 25, 33, 33, 34, 34, 37, 37, 40, 42, 46, 16, 19, 24, 25, 25, 33, 33, 34, 34, 37, 37, 40, 42, 46, 49, 49, 73 73

Find the mean of the original data.Find the mean of the original data.• Enter data into L1Enter data into L1• STAT:CALC:1 –Var StatsSTAT:CALC:1 –Var Stats: L1: L1• , Mean = 35.44, Mean = 35.44

Find the mean without outlier.Find the mean without outlier.• Remove 73 from L1Remove 73 from L1• STAT:CALC:1 –Var StatsSTAT:CALC:1 –Var Stats: L1: L1• Mean = 32.93Mean = 32.93

We can find median also with 1-Var Stats. We can find median also with 1-Var Stats. Scroll down the list.Scroll down the list.

x

Ex: SSHA ScoresEx: SSHA Scores

The Survey of Study Habits and The Survey of Study Habits and Attitudes (SSHA) is a psychological Attitudes (SSHA) is a psychological test that evaluates college students’ test that evaluates college students’ motivation, study habits, and motivation, study habits, and attitudes toward school. A private attitudes toward school. A private college gives the SSHA to a sample college gives the SSHA to a sample of 18 of its incoming first-year of 18 of its incoming first-year women students. Their scores are: women students. Their scores are:

154154 109109 137137 115115 152152 140140 154154 178178101101103103 126126 126126 137137 165165 165165 129129 200200148148

Make a stemplot of these data. Make a stemplot of these data. Enter into L2 and sort, then use: Enter into L2 and sort, then use:

• Stems from 10 -20Stems from 10 -20• Leaves: Leaves: ones; so 154 looks like 15/4ones; so 154 looks like 15/4

Are there any potential outliers?Are there any potential outliers?About where is the center of the About where is the center of the distribution ?distribution ?

Potential outliers:Potential outliers: 1010 139 139

Center: Center: 11 11 5 5Median: mean of the 9Median: mean of the 9thth and 10 and 10thth observ. 12 observ. 12

669669 1313 7 777 1414 0088 1515 244 244 1616 55 55 1717 8 8 1818

1919 2020 0 0

200200

about 138.5about 138.5

n = 18n = 18

ShapeShape

Describe shape:Describe shape:The overall shape of the distribution The overall shape of the distribution is irregular, as often happens when is irregular, as often happens when only a few observations are only a few observations are available. available.

SpreadSpread

What is the spread of the scores What is the spread of the scores (ignoring any outliers)? (ignoring any outliers)?

178 – 101 = 77 or from 101 to 178 178 – 101 = 77 or from 101 to 178

CenterCenter b. : Find the mean score from the b. : Find the mean score from the

formula for the mean.formula for the mean. By hand:By hand:Sum of the 18 observations/18Sum of the 18 observations/18

= 2539/18 == 2539/18 =

Calculator keystrokesCalculator keystrokes::2nd STAT(list):MATH:MEAN(L2)2nd STAT(list):MATH:MEAN(L2) or orSTAT:CALC:1-Var Stat (L2)STAT:CALC:1-Var Stat (L2)

==

x

x

141.06

141.06

c. Find the median of these scores.c. Find the median of these scores. Which is larger: the median or the mean? Which is larger: the median or the mean?

Median = average of the 9Median = average of the 9thth and 10 and 10thth scores scores

= 138.5= 138.5

vs. mean = 141.058vs. mean = 141.058

Explain why? Explain why?

The mean is larger than the median The mean is larger than the median because of the because of the outlieroutlier at 200 which at 200 which pulls pulls the mean toward the long right tail of the mean toward the long right tail of the distribution.the distribution.

Describe the following Describe the following distributions:distributions:

Both distributions are roughly symmetric.

The center for both distributions is 90 goals.

Both distributions have different amounts of VARIABILITY.

Measuring Measuring Spread/VariabilitySpread/Variability

• RangeRange:: The difference between the The difference between the largest and smallest observations.largest and smallest observations.

• QuartilesQuartiles:: (uses median) The (uses median) The quartiles quartiles mark out the middle half mark out the middle half and and improve our description of improve our description of spread.spread.

QuartilesQuartiles

1. Arrange observations in increasing order and locate 1. Arrange observations in increasing order and locate M. M.

2.2. TheThe first quartile (Q1)first quartile (Q1)

• lies one-quarter of the way up list of ordered lies one-quarter of the way up list of ordered observationsobservations

• M of lower half M of lower half

• larger than 25% of ordered observations.larger than 25% of ordered observations.

3. The3. The third quartile (Q3)third quartile (Q3)lies three-quarters of the lies three-quarters of the

way up list of way up list of ordered observationsordered observations

• larger than 75% of larger than 75% of ordered ordered observations.observations.

• M of upper halfM of upper half 4.4. TheThe “second “second

quartile”quartile”

• median (M)median (M)

• Note: is not larger Note: is not larger than 50% of ordered than 50% of ordered observationsobservations

• is at 50% markis at 50% mark

Barry Bonds cont.Barry Bonds cont.

Locate Q1 and Q3:Locate Q1 and Q3:

Q1 = 25Q1 = 25

Q3 = 41Q3 = 41

Interquartile RangeInterquartile Range (IQR)(IQR)::

The distance between the first and The distance between the first and third quartiles.third quartiles.

IQR = Q3 – Q1IQR = Q3 – Q1

IQR = 41 – 25 = 16IQR = 41 – 25 = 16

Note:Note: If an observation falls in If an observation falls in the IQR it is not unusually high the IQR it is not unusually high or low.or low.We use IQR to identify suspect outliers.We use IQR to identify suspect outliers.

OutliersOutliers

• Unusually high or unusually lowUnusually high or unusually low

• Basic “rule of thumb” for identifying Basic “rule of thumb” for identifying is if the observation is if the observation falls more than falls more than 1.5 x IQR 1.5 x IQR above the third quartile above the third quartile or or below the first quartile.below the first quartile.

OutliersOutliers

1.1.Find IQRFind IQR

2.2. Q3 + 1.5 x IQR Q3 + 1.5 x IQR (upper (upper cutoff)cutoff)

3.3. Q1 – 1.5 x IQR Q1 – 1.5 x IQR (lower cutoff)(lower cutoff)

Is Barry bonds 73 an Is Barry bonds 73 an outlier?outlier?• IQR =IQR =

41 – 25 41 – 25 = 16= 16

• 1.5 x IQR = 1.5 x IQR =

= 24= 24

• Q3 + 24 =Q3 + 24 =

41 + 24 = 6541 + 24 = 65 (upper cutoff)(upper cutoff)

• Yes or no?Yes or no?

Yes Bonds record setting year of 73 is an Yes Bonds record setting year of 73 is an outlier.outlier.

Ex: McDonald’s Chicken Ex: McDonald’s Chicken SandwichesSandwichesProblem:Problem: Determine whether the Premium Crispy Determine whether the Premium Crispy

Chicken Club Sandwich with 28 grams of fat is Chicken Club Sandwich with 28 grams of fat is an outlier. (2 min)an outlier. (2 min)

Solution:Solution:

Here are the 14 amounts of fat in order:Here are the 14 amounts of fat in order:

9 9 10 10 12 15 16 16 17 17 17 20 9 9 10 10 12 15 16 16 17 17 17 20 23 2823 28 M1Q 3Q

17 – 10 7 grams of fatIQR

Which is resistant range or Which is resistant range or IQR?IQR?

•Range:Range:

• IQR:IQR:

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