1. Describing distributions with numbers
A large number or numerical methods are available for
describing quantitative data sets. Most of these methods
measure one of two data characteristics:
The central tendency of the set of observations – it is the
tendency of the data to cluster, or center, about certain
numerical values.
The variability of the set of observation – it is the spread
of the data.
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2. Measuring Center
The Mode is the observation that occurs most frequently.
The mode for categorical variable will be the label of the
category with the highest number of counts.
Measuring center
Two common measures of center are the mean and the median.
These two measures behave differently.
The mean is the “average value” and the median is the “middle
value”.
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3. Measuring center: the median
The median M is the midpoint of the distribution, the number
such that half the observations are smaller then it and the
other half are larger.
To find the median of a distribution:
Arrange the observations in order of size, from smallest to largest.
2. If the number of observations n is odd, the median is the
center observation in the ordered list.
3. If the number of observations n is even, the median is the
average of the two center observations in the ordered list.
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4. Example The annual salaries (in thousands of $) of a random
sample of five employees of a company are:
40, 30, 25, 200, 28
Arranging the values in increasing order:
median = 30
Excluding 200 median = (28+30)/2=29.
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5. MINITAB output for the data in the example above is given bellow:
MINITAB commands Stat > Basic Statistics > Display Descriptive Statistics
MINITAB output for the data in the example above is given bellow:
Variable N Median
salary
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6. Measuring center: mean
To find the mean of a set of observations, add their values and divide by the number of observations. If the n observations are x1,x2,…xn, their mean is given by
Example
Find the mean of the following observations: 4, 5, 9, 3, 5.
Solution:
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7. Example
The annual salaries (in thousands of $) of a random sample of five employees of a company are: 40, 30, 25, 200, 28.
If we exclude 200 as an outlier,
Mean is sensitive to the influence of a few extreme
observations. Because the mean cannot resist the influence of
extreme values, we say that it is NOT a resistant measure of
center.
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8. Example - Calculation for grouped data
Determine the mean of the data represented by the following frequency table.
Solution:
Class Interval
Frequency
10-20 2
20-30 4
30-40 7
40-50 6
50-60 1
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9. Mean versus median
The median and mean are the most common measures of the
center of a distribution.
If the distribution is exactly symmetric, the mean and median
are exactly the same.
Median is less influenced by extreme values.
If the distribution is skewed to the right, then
mode < median < mean
If the distribution is skewed to the left, then
mean < median < mode.
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10. Trimmed mean
Trimmed mean is a measure of the center that is more resistant than the mean but uses more of the available information than the median.
To compute the 10% trimmed mean, discard the highest 10% and the lowest 10% of the observations and compute the mean of the remaining 80%. Similarly, we can compute 5%, 20% etc. trimmed mean.
Trimming eliminates the effect of a small number of outliers.
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11. Example Compute the 10% trimmed mean of the data given below.
Solution:
- Arrange the values in increasing order:
- The are 20 observations and 10% of 20 = 2. Hence, discard
the first 2 and the last 2 observations in the ordered data and
compute the mean of the remaining 16 values.
Variable N Mean C
Exercise 1.77 on page 63 in IPS.
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12. Questions You are asked to recommend a measure of center to
characterize the following data:
0.6, 0.2, 0.1, 0.2, 0.2, 0.3, 0.7, 0.1, 0.0, 22.5, 0.4.
What is your recommendation and why?
The mean is ____ sensitive to extreme values than the median.
(a) more
(b) less
(c) equally
(d) can’t say without data
Changing the value of a single score in a data set will necessarily cause the mean to change. (T/F)
Changing the value of a single score in a data set will necessarily cause the median to change. (T/F)
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13. Percentiles
The simplest useful numerical description of a distribution consists of both a measure of center and a measure of spread.
We can describe the spread or variability of a distribution by giving several percentiles.
The pth percentile of a distribution is the value such that p percent of the observations are smaller or equal to it.
The median is the 50th percentile.
If a data set contains n observations, then the pth percentile is the value in the ordered data set.
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14. Example Find the 20th percentile of the data represented by the
following stem-and-leaf plot.
Stem-and-leaf of Rural N = 29
Leaf Unit = N* = 7
(12)
1 9
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15. Solution
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16. Quartiles The 25th percentile is called the first quartile (Q1).
The first quartile (Q1) is the median of the observations whose position in the ordered list is to the left of the location of the overall median.
The 75th percentile is called the third quartile (Q3).
The third quartile (Q3) is the median of the observations whose position in the ordered list is to the right of the location of the overall median.
NOTE: The median is the second quartile Q2 .
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17. Example
The highway mileages of 20 cars, arranged in increasing order are: |
32.
The median is …
The first quartile Q1 is …
The third quartile Q3 is…
Exercise: Find
(a) the 10th percentile.
(b) the 90th percentile of the above data set.
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18. Measuring Spread
The range (max-min) is a measure of spread but it is very sensitive to the influence of extreme values.
The distance between the first and third quartiles is called the Interquartile range (IQR) i.e. IQR =Q3 – Q1 .
The IQR is another measure of spread that is less sensitive to the influence of extreme values.
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19. The five-number summary
The five-number summary of a set of observations consists of the smallest observation, the first quartile, the median, the third quartile and the largest observation.
These five numbers give a reasonably complete description of both the center and the spread of the distribution.
MINITAB commands: Stat > Basic Statistics > Display Descriptive Statistics
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20. Example
The highway mileages of 20 cars, arranged in increasing order are:
Give the five number summary.
Answer
From example 1.14 p45 we have, min = 13, first quartile = 18,
median = 23, third quartile = 27 , max. = 32.
The MINITAB output using the above commands is as follows:
Variable N Minimum Q1 Median Q3 Maximum
mileage
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21. Box-plot A box-plot is a graph of the five-number summary. Example:
Make a box-plot for the data in the above example.
MINITAB commands: Graph > Boxplot
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22. Exercise The dot-plot for a set of 20 observations is given below:
Draw a box-plot for the data (use the same scale).
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23. Exercise The stemplot for a set of 50 observations is given below:
Draw a box-plot for the data.
Compute the 40% trimmed mean.
Stem-and-leaf of Fees N = 50
Leaf Unit = 1.0
(28)
1 4
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24. Exercise
The box-plot, histogram and stem-and-leaf plot for a data set are given below. Describe the distribution.
Stem-and-leaf of C N = 50
Leaf Unit = 1.0
(29)
1 3
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25. Exercise
Consider the following Minitab generated box-plots of coagulation times in seconds for samples of blood drawn from animals receiving three different diets denoted 1, 2, and 3 :
State whether the following statements are true or false
a) The animal that had the longest coagulation time was given diet 3.
b) The greatest variability occurs with diet 2.
c) Diet 1 shows evidence of right (positive) skewness but diet 2 shows evidence
of left (negative) skewness.
d) Approximately 25% of animals on diet 2 had coagulation times less then 63.
e) The smallest upper (third) quartile is for diet 3.
f) We can see that the mean for diet 1 is less than 62 seconds.
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26. Measuring spread: Standard deviation
The variance (s2) of a set of n observations is
The standard deviation (s) is the square root of the variance (s2). i.e.
It can be shown that,
This formula is usually quicker.
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27. The deviations display the spread of the values xi about
their mean. Some of these deviations will be positive and some
negative because the observations fall on each side of the
mean.
The sum of the deviations of the observations from their mean
will always be zero.
Squaring the deviations makes them all positive, so that
observations far from the mean in either direction have large
positive squared deviations.
The variance is the average of the squared deviations.
The variance, s2, and the standard deviation, s, will be large if
the observations are widely spread about their mean, and small
if the observations are all close to the mean.
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28. Example
Find the standard deviation of the following data set: 4, 8, 2, 9, 7.
Solution: n=5 ,
Using the second formula we have
and so
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29. MINITAB commands Stat > Basic Statistics > Display Descriptive Statistics
MINITAB output for the above data is given below:
Variable N StDev C
Exercise: Find the standard deviation of the following data set: 5, 8, 7, 9, 7, 11.
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30. Example -Calculation from grouped data
Determine the standard deviation of the data represented by the
following frequency distribution.
Class Interval
Frequency
10-20 2
20-30 4
30-40 7
40-50 6
50-60 1
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31. Solution
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32. Properties of standard deviation (s)
s measures the spread about the mean and should be used only when the mean is chosen as the measure of center.
s = 0 only when there is no spread. This happens only when all observations have the same value. Otherwise, s > 0.
s, like the mean , is not resistant to extreme values. A few outliers can make s very large.
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33. Ballpark approximation for s
The ballpark approximation for the standard deviation s is the Range/4 (divide by 3 if there are less then 10 observations, divide by 5 if there are more then 100 observations).
For the data set 4, 8, 2, 9, 7, range = 9 – 2 = 7 and so .
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34. The empirical (68-95-99.7) rule
With a bell shaped distribution,
about 68% of the data fall within a distance of 1 standard
deviation from the mean.
95% fall within 2 standard deviations of the mean.
99.7% fall within 3 standard deviations of the mean.
What if the distribution is not bell-shaped?
There is another rule, named Chebyshev's Rule, that tells us
that there must be at least 75% of the data within 2 standard
deviations of the mean, regardless of the shape, and at least
89% within 3 standard deviations.
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35. Outliers An outlier is an observation that is usually large or small
relative to the other values in a data set. Outliers are typically
attributable to one of the following causes:
1. The observation is observed, recorded, or entered incorrectly.
2. The observation comes from a different population.
3. The observation is correct but represents a rare event.
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36. The 1.5×IQR Criterion for outliers
Call an observation a suspected outlier if it falls more than 1.5×IQR above the 3rd quartile or below the 1st quartile.
Example
Consider the data given in example 1.13 on page 43 in IPS (mileage data with an extra observation of 66).
Variable N Mean Min Q1 Median Q3 Max
Mileages
The IQR = = 10 and the largest observation, 66, falls
more than 1.5×IQR above Q3 and therefore is an outlier.
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37. Choosing a summary
The five-number summary is usually better than the mean and the standard deviation for describing skewed distributions or distributions with strong outliers.
Use mean and standard deviation for reasonably symmetric distributions that are free of outliers.
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38. Questions
1. How do the mean, median, and mode compare, usually, when a distribution is positively skewed? negatively skewed? Draw a picture and try to estimate the locations of these measures.
2. Which type of display is the most useful type for clear direct comparisons of the key characteristics of several data sets (e.g. blood cholesterol changes for several different treatments) ?
3. In a frequency table of 300 scores, the mean is reported as 80 and the median as 65. One would expect this distribution to be
a. positively skewed.
b. negatively skewed.
c. symmetrical
d. rectangular.
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39. 4. Find the median of the following frequency distribution.
5. On sta220 term test, John scored at the 78th percentile, and Jack scored at the
63rd. State whether the following statements are true of false
a. John is 15 times better than jack.
b. John scored 15 more points than Jack.
c. 15% of those taking the test got scores ranging between
John's and Jack's scores.
d. 62 students scored less than John.
Score
Frequency 1 2 9 3 5 4
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40. 6. Estimate the mean and standard deviation of the distribution represented by the following histogram.
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