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### Section 1.2

Describing Distributions with Numbers

Quantitative Data

- Measuring Center
- Mean
- Median
- Measuring Spread
- Quartiles
- Five Number Summary
- Standard deviation
- Boxplots

Measures of Center

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

Calculations

- 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

Mean vs. Median

- 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.

Measures of spread

- 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)

Calculating median

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

Calculating Quartiles:

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 =

Five-Number Summary

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

Boxplot

- 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

Find the 5 # summary and make a boxplot

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

10 12 13 20 24 26 27 29 30 32 34 34 38 39 39 40 40 44 44 44 44 45 47

Criterion for suspected outliers

- 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

Criterion for suspected outliers

- Are there any outliers?

Criterion for suspected outliers

- Find 5 number summary:

Min Q1 Median Q3 Max

1 54.5 103.5 200 2631

- 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.

Criterion for suspected outliers

- Seven high outliers circled…

Modified Boxplot

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

Standard deviation

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

DATA: 1792 1666 1362 1614 1460 1867 1439

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

Properties of the standard deviation

- 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.

Find the mean and standard deviation.

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

13 27 26 44 30 39 40 34 45 44 24 32 44 39 29 44 38 47 34 40 20 12 10

Linear Transformations: changing units of measurements

- 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:
- _

Linear Transformations

- 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

Effect of a linear transformation

- 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)

Effects of Linear Transformations

- 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)

Example

- 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 =

SAS tips

- “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

Summary (1.2)

- 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

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