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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 :.

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section 1 2

Section 1.2

Describing Distributions with Numbers

quantitative data
Quantitative Data
  • Measuring Center
    • Mean
    • Median
  • Measuring Spread
    • Quartiles
    • Five Number Summary
    • Standard deviation
  • Boxplots
measures of center
Measures of Center
  • The mean
    • The arithmetic mean of a data set (average value)
    • Denoted by :
calculations
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?
slide6
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
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
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
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
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
Five-Number Summary
  • 5 numbers
    • Minimum
    • Q1
    • Median
    • Q3
    • Maximum
find the 5 number summaries
Find the 5-Number Summaries

Example:

Data: 26 13 35 76 44 58

Example:

Data: 84 89 89 64 78

boxplot
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
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
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 outliers17
Criterion for suspected outliers
  • Are there any outliers?
criterion for suspected outliers18
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 outliers19
Criterion for suspected outliers
  • Seven high outliers circled…
modified boxplot
Modified Boxplot
  • Has outliers as dots or stars.
  • The line extends only to the first non-outlier.
standard deviation
Standard deviation
  • Deviation :
  • Variance : s2
  • Standard Deviation : s
slide22
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
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
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
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
Linear Transformations
  • Do not change shape of distribution
  • However, change center and spread

Example: weights of newly hatched pythons:

slide27
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
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
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
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
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
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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