Ch. 3 Numerically Summarizing
Data • The arithmetic mean of a variable is
computed by determining the sum of all
the values of the variable in the data set
divided by the number of observations.
• The population arithmetic mean is
computed using all the individuals in a
population.
– The population mean is a parameter.
– The population arithmetic mean is denoted by
the symbol μ
Population Mean
• If x1, x2, …, xN are the N observations of a
variable from a population, then the population
mean, µ, is
1 2 Nx x x
N
Sample Mean
• The sample arithmetic mean is computed
using sample data.
• The sample mean is a statistic.
• The sample arithmetic mean is denoted by
x
Sample Mean
• If x1, x2, …, xN are the N observations of a
variable from a sample, then the sample mean
is
x
1 2 nx x xx
n
Sample Problem Computing a Population
Mean and a Sample Mean
• The following data represent the travel times
(in minutes) to work for all seven employees of
a start-up web development company.
23, 36, 23, 18, 5, 26, 43
a. Compute the population mean of this data.
b. Then take a simple random sample of n = 3
employees. Compute the sample mean.
Obtain a second simple random sample of n =
3 employees. Again compute the sample
mean.
EXAMPLE Computing a Population Mean and a Sample
Mean
(b) Obtain a simple random sample of size n = 3 from the
population of seven employees. Use this simple random sample
to determine a sample mean. Find a second simple random
sample and determine the sample mean.
1 2 3 4 5 6 7
23, 36, 23, 18, 5, 26, 43
3-7 © 2010 Pearson Prentice Hall. All rights reserved
Median
• The median of a variable is the value that
lies in the middle of the data when
arranged in ascending order. We use M to
represent the median.
EXAMPLE Computing a Median of a Data Set with an Odd
Number of Observations
The following data represent the travel times (in minutes)
to work for all seven employees of a start-up web
development company.
23, 36, 23, 18, 5, 26, 43
Determine the median of this data.
EXAMPLE Computing a Median of a Data Set with an
Even Number of Observations
Suppose the start-up company hires a new employee.
The travel time of the new employee is 70 minutes.
Determine the mean and median of the “new” data set.
23, 36, 23, 18, 5, 26, 43, 70
3-11
EXAMPLE Computing a Median of a Data Set with an
Even Number of Observations
The following data represent the travel times (in minutes) to
work for all seven employees of a start-up web development
company.
23, 36, 23, 18, 5, 26, 43
Suppose a new employee is hired who has a 130 minute
commute. How does this impact the value of the mean and
median?
3-12
EXAMPLE Computing a Median of a Data Set with an
Even Number of Observations
The following data represent the travel times (in minutes) to
work for all seven employees of a start-up web development
company.
23, 36, 23, 18, 5, 26, 43
Suppose a new employee is hired who has a 130 minute
commute. How does this impact the value of the mean and
median?
Mean before new hire: 24.9 minutes
Median before new hire: 23 minutes
Mean after new hire: 38 minutes
Median after new hire: 24.5 minutes 3-13
Resistance
• A numerical summary of data is said to be
resistant if extreme values (very large or
small) relative to the data do not affect its
value substantially.
EXAMPLE Describing the Shape of the Distribution
The following data represent the asking price of homes
for sale in Lincoln, NE.
Source: http://www.homeseekers.com
79,995 128,950 149,900 189,900
99,899 130,950 151,350 203,950
105,200 131,800 154,900 217,500
111,000 132,300 159,900 260,000
120,000 134,950 163,300 284,900
121,700 135,500 165,000 299,900
125,950 138,500 174,850 309,900
126,900 147,500 180,000 349,900
3-16
Sample Problem
• Find the mean and median. Use the mean
and median to identify the shape of the
distribution. Verify your result by drawing
a histogram of the data.
One-Variable Statistics Nspire
1. Create a list &
spreadsheets page
2. Title column
3. Enter data into
column
4. Create a calculator
page
5. Click Menu
6. 6:Statistics
– 1:Stat Calculations
– 1: One-Variable Stats
𝑥 = mean
Sx = sample S.D.
σx = population S.D.
Record min, Q1,
Median, Q3, Max
Find the mean and median. Use the mean
and median to identify the shape of the
distribution. Verify your result by drawing a
histogram of the data.
The mean asking price is $168,320 and the
median asking price is $148,700. Therefore, we
would conjecture that the distribution is skewed
right.
3-19
350000300000250000200000150000100000
12
10
8
6
4
2
0
Asking Price
Fre
qu
en
cy
Asking Price of Homes in Lincoln, NE
3-20
Mode
• The mode of a variable is the most
frequent observation of the variable that
occurs in the data set.
• If there is no observation that occurs with
the most frequency, we say the data has
no mode.
– The data on the next slide represent the Vice
Presidents of the United States and their state
of birth. Find the mode.
To order food at a McDonald’s Restaurant, one must
choose from multiple lines, while at Wendy’s
Restaurant, one enters a single line. The following
data represent the wait time (in minutes) in line for a
simple random sample of 30 customers at each
restaurant during the lunch hour. For each sample,
answer the following:
(a) What was the mean wait time?
(b) Draw a histogram of each restaurant’s wait time.
(c ) Which restaurant’s wait time appears more
dispersed? Which line would you prefer to wait in?
Why? 3-25
Sample Problem
1.50 0.79 1.01 1.66 0.94 0.67
2.53 1.20 1.46 0.89 0.95 0.90
1.88 2.94 1.40 1.33 1.20 0.84
3.99 1.90 1.00 1.54 0.99 0.35
0.90 1.23 0.92 1.09 1.72 2.00
3.50 0.00 0.38 0.43 1.82 3.04
0.00 0.26 0.14 0.60 2.33 2.54
1.97 0.71 2.22 4.54 0.80 0.50
0.00 0.28 0.44 1.38 0.92 1.17
3.08 2.75 0.36 3.10 2.19 0.23
Wait Time at Wendy’s
Wait Time at McDonald’s
3-