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Section 1.2

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Section 1.2

Describing Distributions with Numbers

- Measuring Center
- Mean
- Median

- Measuring Spread
- Quartiles
- Five Number Summary
- Standard deviation

- Boxplots

- The mean
- The arithmetic mean of a data set (average value)
- Denoted by :

- Mean highway mileage for the 19 2-seaters:
- Average: 25.8 miles/gallon

- Issue here: Honda Insight 68 miles/gallon!
- Exclude it, the mean mileage: only 23.4 mpg
- What does this say about the mean?

- Problem: Mean can be easily influenced by outliers. It is NOT a resistant measure of center. Median
- Median is the midpoint of a distribution.
- Resistant or robust measure of center.
- i.e. not sensitive to extreme observations

- In a symmetric distribution, mean = median
- In a skewed distribution, the mean is further out in the long tail than the median.
- Example: house prices are usually right skewed
- The mean price of existing houses sold in 2000 in Indiana was 176,200. (Mean chases the right tail)
- The median price of these houses was only 139,000.

- Quartiles: Divides data into four parts (with the Median)
- pth percentile – p percent of the observations fall at or below it.
- Median – 50th percentile
- First Quartile (Q1) – 25th percentile (median of the lower half of data)
- Third Quartile (Q3) – 75th percentile (median of the upper half of data)

Always the (n+1)/2 observation from the ordered data

Example:Data: 1 2 3 4 5 6 7 8 9

(n+1)/2 = 5, so median is the 5thobservation

Median = 5

Example:Data: 1 2 3 4 5 6 7 8 9 10

(n+1)/2 = 5.5, so median is the 5.5thobservation

Median = average of 5 and 6 = 5.5

Example:Data: 1 2 3 4 5 6 7 8 9

Median = 5 = “Q2”

Q1 is the median of the lower half =

Q3 is the median of the upper half =

Example:Data: 1 2 3 4 5 6 7 8 9 10

Median = 5.5

Q1 =

Q3 =

- 5 numbers
- Minimum
- Q1
- Median
- Q3
- Maximum

Example:

Data: 26 13 35 76 44 58

Example:

Data: 84 89 89 64 78

- Visual representation of the five-number summary.
- Central box: Q1 to Q3
- Line inside box: Median
- Extended straight lines: lowest to highest observation, except outliers

Numbers of home runs that Hank Aaron hit in each of his 23 years in the Major Leagues:

1012132024262729303234343839394040444444444547

- Interquartile Range (IQR) = Q3 - Q1
- Observation is a suspected outlier IF it is:
- greater than Q3 + 1.5*IQR
OR

- less than Q1 – 1.5*IQR

- greater than Q3 + 1.5*IQR

- Are there any outliers?

- Find 5 number summary:
MinQ1MedianQ3Max

154.5103.52002631

- Are there any outliers?
- Q3 – Q1 = 200 – 54.5 = 145.5
- Times by 1.5: 145.5*1.5 = 218.25

- Add to Q3: 200 + 218.25 = 418.25
- Anything higher is a high outlier 7 obs.

- Subtract from Q1: 54.5 – 218.25 = -163.75
- Anything lower is a low outlier no obs.

- Seven high outliers circled…

- Has outliers as dots or stars.
- The line extends only to the first non-outlier.

- Deviation :
- Variance : s2
- Standard Deviation : s

DATA: 1792166613621614146018671439

Mean = 1600

- Find the deviations from the mean:
Deviation1 = 1792 – 1600 = 192

Deviation2 = 1666 – 1600 = 66

…Deviation7 = 1439 – 1600 = -161

- Square the deviations.
- Add them up and divide the sum by n-1 = 6, this gives you s2.
- Take square root: Standard Deviation = s = 189.24

- Standard deviation is always non-negative
- s = 0 when there is no spread
- s is not resistant to presence of outliers
- 5-number summary usually better describes a skewed distribution or a distribution with outliers.

- s is used when we use the mean
- Mean and standard deviation are usually used for reasonably symmetric distributions without outliers.

Numbers of home runs that Hank Aaron hit in each of his 23 years in the Major Leagues:

1327264430394034454424324439294438473440201210

- xnew = a + bxold
- Common conversions
- Distance: 100km is equivalent to 62 miles
- xmiles = 0 + 0.62xkm

- Weight: 1ounce is equivalent to 28.35 grams
- xg= 0 + 28.35 xoz ,

- Temperature:
- _

- Distance: 100km is equivalent to 62 miles

- Do not change shape of distribution
- However, change center and spread
Example: weights of newly hatched pythons:

- Ounces
- Mean weight = (1.13+…+1.16)/5 = 1.12 oz
- Standard deviation = 0.084

- Grams
- Mean weight =(32+…+33)/5 = 31.8 g
- or 1.12 * 28.35 = 31.8

- Standard deviation = 2.38
- or 28.35 * 0.084 = 2.38

- Mean weight =(32+…+33)/5 = 31.8 g

- Multiplying each observation by a positive number b multiplies both measures of center (mean and median) and measures of spread (IQR and standard deviation) by b.
- Adding the same number a to each observation adds a to measures of center and to quartiles and other percentiles but does not change measures of spread (IQR and standard deviation)

- Your Transformation: xnew = a + b*xold
- meannew = a + b*mean
- mediannew = a + b*median
- stdevnew = |b|*stdev
- IQRnew = |b|*IQR
|b|= absolute value of b (value without sign)

- Winter temperature recorded in Fahrenheit
- mean = 20
- stdev = 10
- median = 22
- IQR = 11

- Convert into Celsius:
- mean = -160/9 + 5/9 * 20 = -6.67 C
- stdev = 5/9 * 10 = 5.56
- median =
- IQR =

- “proc univariate” procedure generates all the descriptive summaries.
- For the time being, draw boxplots by hand from the 5-number summary
- Optional: proc boxplot.

- See plot.doc

- Measures of location: Mean, Median, Quartiles
- Measures of spread: stdev, IQR
- Mean, stdev
- affected by extreme observations

- Median, IQR
- robust to extreme observations

- Five number summary and boxplot
- Linear Transformations