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Chapter 3 (continued)
Nutan S. Mishra
Exercises 3.11-3.15
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Size of the data set = 12 for all the five problems
In 3.11 variable x1 = monthly rent of an apartment ($)
In 3.12 variable x2 = monthly phone bill ($)
In 3.13 variable x3 = price of gasoline ($/gallon)
In 3.14 variable x4 = amount paid to doctor ($/month)
In 3.15 variable x5 = prices of beer in a city ($)
More description of the variables is given on page 82
Since all these five variables describe the amount of
money, they are all continuous variables. They may take
values between 0 to infinity.
Note
Most of the statistical software including
minitab prefer the raw (ungrouped) data as
input and not the grouped data.
Shapes of frequency distributions
• Bell-shaped
A bell-shaped picture, shown here,
usually represents a normal
distribution
• Bimodal
A bimodal shape, shown here, has
two peaks. This shape may show
that the data has come from two
different systems. If this shape
occurs, the two sources should be
separated and analyzed separately.
Shapes of frequency distributions
Some histograms will show a skewed distribution to
the right. A distribution skewed to the right is said to
be positively skewed. This kind of distribution has a
large number of occurrences in the lower value cells
(left side) and few in the upper value cells (right side).
Some histograms will show a skewed distribution
to the left, as shown below. A distribution skewed
to the left is said to be negatively skewed. This
kind of distribution has a large number of
occurrences in the upper value cells (right side)
and few in the lower value cells (left side).
Parameters and Statistics
• Values of different numerical measures for
population are called population parameters. For
example population mean µ and population
standard deviation σ are population parameters
• Values of different numerical measures for
sample are called sample statistics. For example
sample mean and sample standard deviation s
are sample statistics.
• When the population is very very large, (most
often) the population parameters are unknown
and then we use sample statistics instead.
Interpreting the Standard Deviation
• Given two samples from a population, the
sample with the larger standard deviation
(SD) is the more variable
– Example :
sx  21.4; s y  29.6
• We are using the SD as a relative or
comparative measure
• How does the SD provide a measure of
variability for a single sample or, what does
29.6 really mean?
Interpreting the Standard Deviation
(continued)
• Consider the list of numbers:
10, 20, 30, 45, 50, 70, 85, 90
• How many measurements are within 1
SD, 2 SDs of the mean?
y  s  50  29.6  20.4
y  s  50  29.6  79.6
For  1 SD 4 out
of 8, or 50%
y  2s  50   2  29.6   9.2 For  2 SD 8 out
y  2s  50   2  29.6   109.2
of 8, or 100%
Chebyshev’s Rule
• Applies to any data set, regardless of
the shape of its frequency distribution
• No useful information on fraction of
measurements falling within  y  s, y  s 
for samples and    ,    for
populations
• At least 3 4 of the measurements will fall
w/in 2 SD of the mean; at least 8 9 of the
measurements will fall w/in 3 SD of the
mean
Chebyshev’s Rule (continued)
• General formulation:
For any number k  1, at least 1  12 of the
k
measurements will fall within k SDs of the
mean  y  ks, y  ks  for samples
   k ,  k   for populations
• Gives the smallest percentages that are
mathematically possible; the observed
percentages can be much higher
The Empirical Rule
A rule of thumb that applies to data sets
that have a bell shaped, symmetric
distribution
–Approximately 68% of the measurements
will fall within 1 SD of the mean
–Approximately 95% of the measurements
will fall within 2 SDs of the mean
–Approximately 99.7% of the measurements
will fall within 3 SDs of the mean
Solution to 3.78(a)
• Variable x = time taken to complete the race by a participant
• Given µ = 220 minutes σ = 20 minutes
• To find the percentage of people who completed their race
between 180 and 260 minutes.
40
40
180
220
260
thus the numbers 180 and 260 are equi distant from the
mean.
In terms of σ, 40 = k σ i.e. 40 = k 20 i.e. k = 2
That is 180 and 260 are at a distance 2σ from mean
Then by Chebyshev’s theorem at least (1 – ¼)% of runners
completed the race between 180 and 260 minutes.
Solution to 3.83 (a)
• Variable x = annual salary of a teacher assistant in the
state of Connecticut
• Given that µ = 24,317 σ = 2000
• To find the percentage of the teacher assistants in the
state whose annual salary is between 20,317 and 28,317
• Also given that salary distribution has bell shaped curve.
• Let us compute the distance between mean and 20,317
in terms of σ :
• 24,317 – 20,317 = 4000 = 2 σ
• Similarly 28,317 -24,317 = 4000 = 2 σ
• Thus using Empirical rule approximately 95% teacher
assistants earn between the given two numbers.
Quartiles
Are the values of variable x those divide the
ordered dataset into four equal parts.
There are three quartiles which divide an ordered
data set into four equal parts; Q1. Q2, Q3
Q1
Q2
Q3
Obviously Q2 is the value which divides dataset into
two equal parts thus Q2 is the median
Q3 – Q1 is called inter quartile range.
Examples of quartiles
N = 15 (N odd)
Original Data
55 36 98 5 56 62 55 77 41 56 56 50 58 81 55
Ordered Data
5 36 41 50 55 55 55 56 56 56 58 62 77 81 98
Quartile Positions . .
. Q1 . . .
M..
Q3 . . .
Positions
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
N = 16 (N even)
Original Data 50 49 91 82 32 49 51 46 74 56 98 50 49 5 59 88
Ordered Data 5 32 46 49 49 49 50 50 51 56 59 74 82 88 91 98
Quartile positions . . . Q1 . . .
-MED- . . .
Q3 . . .
Positions
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Median at average the two middle positions when N is even.
Box plots
• To draw a box plot for the given dataset
we need five summary measures
• Max value, Min value and three quartiles
• Inner fences = 1.5 * inter quartile range
• We will draw box plots with the help of
minitab.