Descriptive Statistics Part II

1
Chapter 3

• Descriptive Statistics Part II
• Describing Central Tendency
• Measures of Variation

2
Important Characteristics of data

• Center A value that indicates where the middle
  of the data set is located.
• Variation A measure of the amount that the data
  values vary among themselves
• Distribution The nature or shape of the
  distribution of data (such as bell-shaped,
  uniform, or skewed)
• Outliers Sample values that lie very far away
  from the vast majority of the other samples

3
Describing Central Tendency

• Mean, ?, is the average or expected value
• Median, Md , is the middle point of the ordered
  measurements
• Mode, Mo, is the most frequent value
• Percentiles and Quartiles

4
Basic Symbols

• The sample size, i.e. the number of items in the
  sample, is denoted n.
• The population size, i.e. the total number of
  items in the entire population, is denoted N.

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Mean

• If the data are from a population, the mean is
  denoted by ? (mu).
• If the data are from a sample, the mean is
  denoted by .
• The sample mean is a point estimate of the
  population mean .

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Example 3.1

• Suppose we compiled a sample of the weights of 5
  professional football players
• 255, 216, 346, 300, 270

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Example 3.2Given below is a sample of monthly
rent values () for one-bedroom apartments. The
data is a sample of 70 apartments in a particular
city. The data are presented in ascending order.

Anderson, Sweeney, and Williams
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The Median

• The median is a value such that at least 50
  of all measurements are less than or equal to it
  and at least 50 of all measurements are greater
  than or equal to it .
• The median is the measure of location most often
  reported for annual income and property value
  data.
• This measure is used instead of the mean since a
  few extremely large incomes or property values
  can inflate the mean.

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• The median Md is found as follows
• Arrange values in ascending order (smallest to
  largest).
• If the number of measurements is odd, the median
  is the middle value.
• If the number of measurements is even, the median
  is the average of the two middle values.

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Example 3.3

• Suppose the following represent a sample of
  salaries of 13 Internist(x1000)
• 127 132 138 141 144 146 152 154
  165 171 177 192 241
• Since n 13 (odd,) then the median is the
  middlemost or 7th measurement, Md152

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Example 3.2 Revisited
Median (475 475)/2 475
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Mode
The mode, Mo , is the measurement that occurs
most frequently.

• The greatest frequency can occur at two or more
  different values.
• If the data have exactly two modes, the data are
  bimodal.
• If the data have more than two modes, the data
  are multimodal.
• Mode is an important measure of location for
  qualitative data (can not compute median and mean
  for qualitative data)

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Mode
450 occurred most frequently (7 times) Mode
450
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Percentiles and Quartiles

• A percentile provides information about how the
  data are spread over the interval from the
  smallest value to the largest value.
• Admission test scores for colleges and
  universities are frequently reported in terms of
  percentiles.

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Percentiles

• The pth percentile of a data set is a value such
  that at least p percent of the items take on this
  value or less and at least (100 - p) percent of
  the items take on this value or more.
• Steps for computing percentiles
• Arrange the data in ascending order.
• 2. Compute index i, the position of the pth
  percentile. i (p/100)n
• 3a. If i is not an integer, round up. The p th
  percentile is the value in this position.
• 3b. If i is an integer, the p th percentile is
  the average of the values in positions i and i 1.

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Example 3.2 Revisited
90th Percentile

• i (p/100)n (90/100)70 63
• Averaging the 63rd and 64th data values
• 90th Percentile (580 590)/2 585

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65th Percentile
i (p/100)n (65/100)70 45.5 This is a non
integer so round i up to 46. Data value in
position 46500 65th Percentile 500
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Quartiles

• Quartiles are specific percentiles
• First Quartile 25th Percentile
• Second Quartile 50th Percentile Median
• Third Quartile 75th Percentile

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Example 2.4 Revisited

• Third quartile 75th percentile
• i (p/100)n (75/100)70 52.5 53
• Third quartile 525

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Measures of Variation

• The range is the largest minus the smallest
  measurement.
• The variance is the average of the sum of the
  square of the deviations from the mean.
• The standard deviation is the square root of the
  variance.
• In a comparison of multiple variables, the one
  with the largest variance shows the most
  variability in the data.

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Example 3.3 Revisited
Internists Salaries (in thousands of dollars)
127 132 138 141 144 146 152 154 165 171 177 192
241 Range 241 - 127 114 (114,000)
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Example 3.2 Revisited
Range largest value - smallest value Range
615 - 425 190
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Variance
If the data set is a sample, the variance is
denoted by s2.
If the data set is a population, the variance is
denoted by ? 2.
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Standard Deviation
If the data set is a sample, the standard
deviation is denoted s.
If the data set is a population, the standard
deviation is denoted ? (sigma).
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Example 3.1 Revisited (recall 277.4)
xi x I - (xi - )2 x1 255
277.4 x1- 255 - 277.4 -22.4 (x1
- )2 (-22.4)2 501.76 x2 216 277.4
x2 - 216 - 277.4 -61.4 (x2 -
)2 (-61.4)2 3769.96 x3 346 277.4 x3 -
346 - 277.4 68.6 (x3 - )2
(68.6)2 4705.96 x4 300 277.4 x4 -
300 - 277.4 22.6 (x4 - )2
(22.6)2 510.76 x5270 277.4 x5 -
270 - 277.4 -7.4 (x5 - )2
(-7.4)2 54 .76 sum 9543.2
Since this is sample data
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Example 3.1 Revisited (continued)
If this was data from the entire population
xi ? xi-? (xi - ?)2 255 277.4
-22.4 501.76 216 277.4 -61.4 3769.96 346 277.4
68.6 4705.96 300 277.4
22.6 510.76 270 277.4 -7.4 54.76
9543.2
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Example 3.4
Compute the standard deviation of the following
sample data 4, 5, 1, -2, 7
xi xi- (xi- )2 4 3 1 1 5 3 2 4 1 3 -2
4 -2 3 -5 25 7 3 4 16
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Z score

• The z-score is often called the standardized
  value.
• It is a measure of location that tells how far a
  particular observation is from the mean.
• It denotes the number of standard deviations a
  data value xi is from the mean.
• A data value less than the sample mean will have
  a z-score less than zero.
• A data value greater than the sample mean will
  have a z-score greater than zero.
• A data value equal to the sample mean will have a
  z-score of zero.

xi is the data value for which you want the z
score
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Example 3.1 Revisited
255, 216, 346, 300, 270
The z score for the data value 216 is
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Chebyshevs Rule

• Chebyshevs rule applies to any data set,
  regardless of the shape of the distribution of
  the data
• Can be used to make statements about the
  proportion of data values that must be within a
  specified number of standard deviations from the
  mean

• Chebyshevs Rule
• At least (1 - 1/k2) of the items in any data
  set will be within k standard
• deviations of the mean, where k is any
  value greater than 1.
• Implications
• At least 75 of the items must be within k 2
  standard deviations
• of the mean. (i.e. within the interval -
  2s, 2s)
• At least 89 of the items must be within k 3
  standard deviations of the mean. (i.e. within the
  interval - 3s, 3s)
• At least 94 of the items must be within k 4
  standard deviations of the mean. (i.e. within the
  interval - 4s, 4s)

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Example 3.1 Revisited
Let k 1.5 with 277.4 and s
48.84 According to Chebyshevs Rule, At least (1
- 1/(1.5)2) 1 - 0.44 0.56 or 56 football
players weights are between - k(s) 277.4
- 1.5(48.84) 204.14 and
k(s) 277.4 1.5(48.84) 350.66
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Empirical Rule for Normal Populations

• For a Normal distribution, the Empirical rule can
  be used to make statements about the proportion
  of data values that must be within a specified
  number of standard deviations from the mean
• If a population has mean ? and standard
  deviation ? and is described by a normal curve,
  then
• 68.26 of the population measurements lie within
  one standard deviation of the mean ? -?, ? ?
• 95.44 of the population measurements lie within
  two standard deviations of the mean ? -2?, ?
  2?
• 99.73 of the population measurements lie within
  three standard deviations of the mean ? -3?, ?
  3?

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Outliers

• Outliers are defined as sample values that lie
  very far away from the vast majority of the other
  samples
• Your book uses the term unusual to refer to
  outliers
• We will use the course notes to numerically
  determine an outlier or an unusual value, NOT the
  criteria set in your text book.
• If we assume the data is normally distributed,
  there are two ways to numerically determine if a
  sample value is an outlier
• 1.) Determine the interval ( - 3s,
  3s). If a value is OUTSIDE the interval, then
  this value in an outlier.
• 2.) Compute the z-value for a sample value. If
  z gt 3 or z lt -3, then the value is an outlier.

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Example (Outliers)

• For the first quiz in MATH 123, the average quiz
  score was 16 (out of 20 pts.), with a variance of
  4 pts. Would a score of 11 be considered an
  usually low score?
• We were given
• 16, s2 and xi11
• Note s2 not 4 because we must take the square
  root of the variance to get the standard
  deviation
• Using the 1st method we determine the range
• ( - 3s, 3s) (16-3(2)
  , 163(2) (10, 22).
• Since 11 is within this range, this score is not
  unusually low.
• Using the 2nd method we determine the z-value
• Since -2.5 is not less than -3, this score is not
  unusually low.
• Note Either method can be used to determine
  whether a value is an outlier, because both will
  ALWAYS yield the same result.

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The End