Chapter 3 Descriptive Statistics / Describing Distributions with
Numbers
A.
Common Measurements of Location
these measurements give you a sense of where a data point falls in line relative to other
data points.
Note:
If you calculate a stat from a sample it is called a sample statistic.
If you calculate
it from a population statistic then it is called a population statistic.
1.
Mean
– this is the most common measurement.
It is simply the average.
It is one of
three measures of centrality.
a. sample mean

_
x
= ∑x
i
/ n = x
1
+ x
2
+ …. + x
n
b. population mean
– μ = ∑x
i
/ N
note:
n = total number in sample
N = total number in population
2. Median
–M
d
– The middle value when the data is arranged in ascending order; second
measure of center.
 if the data has an odd number of points, then there is a true middle
if the data has an even number of points, then take the average of the two middle points.
Example: If you are given 3, 6, 7, 8, 8, 10
Since there are an even number of points we take (x
3
+ x
4
) / 2 = (7 + 8) /2 = 7.5
3.
Mode
–M
o
 this is the value that occurs with the greatest frequency.
If there is more
than on value that takes on the greatest frequency then we say that the value is bi, tri, or
multimodal; last measure or centrality.
4. Percentiles/Quartiles
– tells us how data are spread over a 100 percent interval from
smallest to largest.
How to calculate percentage:
(a) arrange data in ascending order
(b) Compute the index of the percent
Index – i = (p/100)n
p = percentile of interest
n = number of operations
(c) If it is not an integer round up.
If it is an integer then the pth percentile is average of
the i and i+1 value
a. For Lower Quartile (Q1 or 25
th
percentile):
i.
Sort all observations in ascending order
1
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Compute the position L1 = 0.25 * N, where N is the total number of observations.
iii.
If L1 is a whole number, the lower quartile is midway between the L1th value and
the next one.
iv.
If L1 is not a whole number, change it by rounding up to the nearest integer. The
value at that position is the lower quartile.
b. For Upper Quartile (Q3 or 75
th
percentile):
i.
Sort all observations in ascending order
ii.
Compute the position L3 = 0.75 * N, where N is the total number of observations.
iii.
If L3 is a whole number, the lower quartile is midway between the L3th value and
the next one.
iv.
If L3 is not a whole number, change it by rounding up to the nearest integer. The
value at that position is the lower quartile.
Example: 61, 61, 61, 67, 73, 73, 74, 79, 81, 81, 87, 89, 89, 92, 97, 100
Given our data from test scores again if we wanted to know the 40
th
percentile we obtain
it as follows.
i = (40/100) 16 = .4*16 = 6.4 = 7
So our 7
th
value is our 40
th
percentile.
This corresponds to 74.
5. Quartiles
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 Spring '10
 smith
 Accounting, Standard Deviation, Quartile

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