Download Descriptive Statistics: Numerical Methods

Descriptive Statistics: Numerical Methods,
Part 1
Measures of Location
The Mean
The Median
The Mode
Percentiles
Quartiles
Mean
The mean (or average) is
the basic measure of
location or “central
tendency” of the data.
•The sample mean
sample statistic.
x
is a
•The population mean  is a
population statistic.
Sample Mean
 xi
x
n
Where the numerator is the sum of values of n
observations, or:
 xi  x1  x2  ...  xn
The Greek letter Σ is the summation sign
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:
 xi x1  x2  x3  x4  x5 46  54  42  46  32
x


 44
n
5
5
The average class size is 44 students
Population Mean ()
 xi

N
The number of observations
in the population is denoted
by the upper case N.
The sample mean x is
a point estimator of
the population mean 
Median
The median is the value in the
middle when the data are arranged in
ascending order (from smallest value
to largest value).
a. For an odd number of observations the median
is the middle value.
b. For an even number of observations the
median is the average of the two middle values.
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
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
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
2890  2920
Median 
 2905
2
Mode
The mode is the value that occurs with
greatest frequency
Soft Drink Example
Soft Drink
Frequency
Coke Classic
19
Diet Coke
8
Dr. Pepper
5
Pepsi Cola
13
Sprite
5
Total
50
The mode is Coke
Classic. A mean or
median is
meaningless of
qualitative data
Using Excel to Compute the Mean, Median,
and Mode
Enter the data into cells A1:B13 for the starting salary
example.
•To compute the mean, activate an empty cell and enter
the following in the formula bar:
=Average(b2:b13) and click the green checkmark.
•To compute the median, activate an empty cell and enter
the following in the formula bar:
= Median(b2:b13) and click the green checkmark.
•To compute the mode, activate an empty cell and enter
the following in the formula bar:
=Average(b2:b13) and click the green checkmark.
The Starting Salary Example
Mean
2940
Median
Mode
2905
2880
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
Calculating the pth Percentile
•Step 1: Arrange the data in ascendingorder
(smallest value to largest value).
•Step 2: Compute an index i
 p 
i
n
 100 
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
Example: Starting Salaries of Business Grads
Let’s 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: i   p n   85 12  10.2
 100 
 100 
Step 3: Since 10.2 in not an integer, round up to
11.The 85thpercentile is the 11th position (3130)
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
Let’s compute the 1st and
3rd percentiles using the
starting salary data. Note we
already computed the
median for this sample—so
we know the 2nd quartile
2710 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 3325
Now find the 25th percentile:
 p 
 25 
i
n  
12  3
 100 
 100 
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
 p 
 75 
n  
12  9
 100 
 100 
Now find the 75th percentile: i  
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
Quartiles for the Starting Salary Data
2710 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 3325
Q1 = 2865
Q1 = 2905
(Median)
Q3 = 3000
Using Excel to Compute Percentiles and
Quartiles
Enter Data: Labels and starting salary data are entered into cells
A1:B13
•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 selected>
Click OK