Descriptive Statistics: Numerical Methods, Part 1

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Descriptive Statistics Numerical Methods, Part 1

• Measures of Location
• The Mean
• The Median
• The Mode
• Percentiles
• Quartiles

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Mean
The mean (or average) is the basic measure of
location or central tendency of the data.

• The sample mean is a sample statistic.
• The population mean ? is a population statistic.

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Sample Mean
Where the numerator is the sum of values of n
observations, or
The Greek letter S is the summation sign
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Example College Class Size
We have the following sample of data for 5
college classes 46 54 42 46 32
We use the notation x1, x2, x3, x4, and x5 to
represent the number of students in each of the 5
classes
X1 46 x2 54 x3 42 x4 46 x5
32
Thus we have
The average class size is 44 students
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Population Mean (?)
The number of observations in the population is
denoted by the upper case N.
The sample mean is a point estimator of the
population mean ?
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Median
The median is the value in the middle when the
data are arranged in ascending order (from
smallest value to largest value).

1. For an odd number of observations the median is
   the middle value.
2. For an even number of observations the median is
   the average of the two middle values.

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The College Class Size example
First, arrange the data in ascending order
32 42 46 46 54
Notice than n 5, an odd number. Thus the median
is given by the middle value.
32 42 46 46 54
The median class size is 46
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Median Starting Salary For a Sample of 12
Business School Graduates
A college placement office has obtained the
following data for 12 recent graduates
Graduate Starting Salary Graduate Starting Salary
1 2850 7 2890
2 2950 8 3130
3 3050 9 2940
4 2880 10 3325
5 2755 11 2920
6 2710 12 2880
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First we arrange the data in ascending order
2710 2755 2850 2880 2880 2890 2920 2940
2950 3050 3130 3325
Notice that n 12, an even number. Thus we take
an average of the middle 2 observations
2710 2755 2850 2880 2880 2890 2920 2940
2950 3050 3130 3325
Middle two values
Thus
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Mode
The mode is the value that occurs with
greatest frequency
The mode is Coke Classic. A mean or median is
meaningless of qualitative data
Soft Drink Example
Soft Drink Frequency
Coke Classic 19
Diet Coke 8
Dr. Pepper 5
Pepsi Cola 13
Sprite 5
Total 50
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Using Excel to Compute the Mean, Median, and Mode

• Enter the data into cells A1B13 for the starting
  salary example.
• To compute the mean, activate an empty cell and
  enter the following in the formula
  barAverage(b2b13) and click the green
  checkmark.
• To compute the median, activate an empty cell and
  enter the following in the formula bar
  Median(b2b13) and click the green checkmark.
• To compute the mode, activate an empty cell and
  enter the following in the formula
  barAverage(b2b13) and click the green
  checkmark.

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The Starting Salary Example
Mean 2940
Median 2905
Mode 2880
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Percentiles
The pth percentile is a value such that at
least p percent of the observations are less than
or equal to this value and at least (100 p)
percent of the observations are greater than or
equal to this value.
I scored in the 70th percentile on the Graduate
Record Exam (GRE)meaning I scored higher than 70
percent of those who took the exam
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Calculating the pth Percentile

• Step 1 Arrange the data in ascendingorder
  (smallest value to largest value).
• Step 2 Compute an index i

where p is the percentile of interest and n in
the number of observations.

• Step 3 (a) If i is not an integer, round up. The
  next integer greater than i denotes the position
  of the pth percentile. (b) If i is an integer,
  the pth percentile is the average of values in i
  and i 1

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Example Starting Salaries of Business Grads
Lets compute the 85th percentile using the
starting salary data. First arrange the data in
ascending order.
Step 1
2710 2755 2850 2880 2880 2890 2920
2940 2950 3050 3130 3325
Step 2
Step 3 Since 10.2 in not an integer, round up to
11.The 85thpercentile is the 11th position
(3130)
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Quartiles
Quartiles are just specific percentiles
Let Q1 first quartile, or 25th percentile Q2
second quartile, or 50th percentile (also the
median) Q3 third quartile, or 75th percentile
Lets compute the 1st and 3rd percentiles using
the starting salary data. Note we already
computed the median for this sampleso we know
the 2nd quartile
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2710 2755 2850 2880 2880 2890 2920
2940 2950 3050 3130 3325
Now find the 25th percentile
Note that 3 is an integer, so to find the 25th
percentile we must average together the 3rd and
4th values Q1 (2850
2880)/2 2865
Now find the 75th percentile
Note that 9 is an integer, so to find the 75th
percentile we must average together the 9th and
10th values Q1 (2950
3050)/2 3000
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Quartiles for the Starting Salary Data
2710 2755 2850 2880 2880 2890 2920
2940 2950 3050 3130 3325
Q1 2865
Q3 3000
Q1 2905 (Median)
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Using Excel to Compute Percentiles and Quartiles
Enter Data Labels and starting salary data are
entered into cells A1B13

• Step 1 Activate any cell containing data in
  column B.
• Step 2 Select the Data menu
• Step 3 When the Sort dialog box appears Sort
  by box, make sure that Starting Salary
  appears andthat Ascending is selectedgt Click
  OK