Transcript Chapter Three Numerically Summarizing Data
Chapter 3 Numerically Summarizing Data
3.1
Measures of Central Tendency 1
The following chart gives a summary of some background information on 5 students at Joliet Junior College (JJC).
Name Age Gender Number of Semesters Overall GPA Completed At JJC Jennifer 21 Amy 19 Brian 18 Mark 18 Jim 24 Female Female Male Male Male 1 2 4 2 3 3.5
2.75
3.25
3.0
4.0
Which of the above data would be qualitative?
Answer:Gender 2
A
parameter
a population.
is a descriptive measure of A
statistic
sample.
is a descriptive measure of a A statistic is an
unbiased estimator
of a parameter if it does not consistently over- or underestimate the parameter .
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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.
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The
population arithmetic mean
, is computed using
all
the individuals in a population.
The population mean is a
parameter
.
by 5
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The
sample arithmetic mean
, is computed using sample data.
The sample mean is a statistic that is an unbiased estimator of the population mean
.
The sample arithmetic mean is denoted by
x
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EXAMPLE
Computing a Population Mean and a Sample Mean
The following chart gives a summary of some background information on a Calculus class. Name Age Gender Jennifer 21 Female Amy 19 Female Overall GPA 3.5
2.75
Brian Jane 25 19 Mark 18 Julie 19 Male Female Male Female 3.95
2.75
3.0
3.85
Jim 24 Male 4.0
Ted 25 Male 3.7
Michel 19 Amanda 19 Linda 19 Female Female Female 3.75
3.65
4.0
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Compute the Arithmetic Mean
Treat the students in this class as a population. Compute the population mean of the GPA. Then take a simple random sample of
n
= 5 students. Compute the sample mean of the GPA. Obtain a second simple random sample of
n
= 5 students. Again compute the sample mean of the GPA.
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The population(size of 11) mean
3.5+2.75+ 3.95+2.75+ 3.0+ 3.85+ 4.0
+ 3.7+ 3.75+ 3.65+ 4.0=34.9
X
34 .
9 11 3 .
54 11
The
median
of a variable is the value that lies in the middle of the data when arranged in ascending order. That is, half the data is below the median and half the data is above the median. We use
M
to represent the median.
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EXAMPLE
Computing the Median of Data
Find the population median of the total GPA from the earlier example.
2.75 2.75 3.0 3.5 3.95 3.65
1 2 3 4 5 6 3.7 3.75 3.85 4.0 4.0
7 8 9 10 11 11 1 2 6 14
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
.
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EXAMPLE Finding the Mode of a Data Set The data on the next slide represent the Vice Presidents of the United States and their state of birth. Find the mode.
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The mode is New York .
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The arithmetic mean is sensitive to extreme (very large or small) values in the data set, while the median is not. We say the median is
resistant
to extreme values, but the arithmetic mean is not. 20
When data sets have unusually large or small values relative to the entire set of data or when the distribution of the data is skewed, the median is the preferred measure of central tendency over the arithmetic mean because it is more representative of the typical observation. 21
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EXAMPLE
Identifying the Shape of the Distribution Based on the Mean and Median
The following data represent the asking price of homes for sale in Lincoln, NE.
Source: http://www.homeseekers.com
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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.
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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.
Using MINITAB/Excel/Spss, we find that the mean asking price is $143,509 and the median asking price is $131,825. Therefore, we would conjecture that the distribution is skewed right.
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3.2
Measures of Dispersion 31
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?
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Wait Time at Wendy ’ s 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
Wait Time at McDonald ’ s 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
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The mean wait time in each line is 1.39 minutes.
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The
range, R,
of a variable is the difference between the largest data value and the smallest data values. That is Range =
R
= Largest Data Value – Smallest Data Value 36
EXAMPLE
Finding the Range of a Set of Data
Find the range of the student GPA collected from Section 3.1
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The
population variance
of a variable is the sum of squared deviations about the population mean divided by the number of observations in the population, N.
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The
population variance
is symbolically represented by lower case Greek sigma squared.
Note:
When using the above formula, do not round until the last computation. Use as many decimals as allowed by your calculator in order to avoid round off errors.
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EXAMPLE
Computing a Population Variance
Compute the population variance of the population data collected in Section 3.1.
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The
sample variance
is computed by determining the sum of squared deviations about the sample mean and then dividing this result by n – 1.
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Note:
Whenever a statistic consistently overestimates or underestimates a parameter, it is called
biased
. To obtain an
unbiased
estimate of the population variance, we divide the sum of the squared deviations about the mean by n - 1.
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EXAMPLE
Computing a Sample Variance
Compute the sample variance using the sample data from Section 3.1
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The
population standard deviation
is denoted by It is obtained by taking the square root of the population variance, so that 44
EXAMPLE
Computing a Population Standard Deviation and Sample Standard Deviation
Compute the population and sample standard deviation for the data obtained in Section 3.1
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EXAMPLE
Comparing Standard Deviations
Determine the standard deviation waiting time for Wendy ’ s and McDonald ’ s. Which is larger? Why?
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EXAMPLE
Comparing Standard Deviations
Determine the standard deviation waiting time for Wendy ’ s and McDonald ’ s. Which is larger? Why?
Sample standard deviation for Wendy ’ s: 0.738 minutes Sample standard deviation for McDonald ’ s: 1.265 minutes 47
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EXAMPLE
Using the Empirical Rule
The following data represent the serum HDL cholesterol of the 54 female patients of a family doctor.
41 62 67 60 54 45 48 75 69 60 54 47 43 77 69 60 55 47 38 58 70 61 56 48 35 82 65 62 56 48 37 39 72 63 56 50 44 85 74 64 57 52 44 55 74 64 58 52 44 54 74 64 59 53 50
(a) Compute the population mean and standard deviation.
(b) Draw a histogram to verify the data is bell shaped.
(c) Determine the percentage of patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule. (d) Determine the percentage of patients that have serum HDL between 34 and 80.8 according to the Empirical Rule. (e) Determine the actual percentage of patients that have serum HDL between 34 and 80.8.
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(a) Using a TI83 plus graphing calculator, we find 57 .
4 and 11 .
7 (b) 52
57 .
4 and 11 .
7 (c) According to the Empirical Rule, approximately 99.7% of the patients will have serum HDL cholesterol levels within 3 standard deviations of the mean. That is, approximately 99.7% of the patients will have serum HDL cholesterol levels greater than or equal to 57.4 - 3(11.7) = 22.3 and less than or equal to 57.4 + 3(11.7) = 92.5.
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57 .
4 and 11 .
7 (d) Because 33.8 is 2 standard deviations below the mean (57.4 - 2(11.7) = 34) and 81 is 2 standard deviations above the mean (57.4 + 2(11.7) = 80.8), the Empirical Rule states that approximately 95% of the data will lie between 34 and 80.8.
(e) There are no observations below 34. There are 2 observations greater than 80.8. Therefore, 52/54 = 96.3% of the data lie between 34 and 80.8.
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EXAMPLE
Using Chebyshev’s Theorem
Using the data from the previous example, use Chebyshev’s Theorem to (a) determine the percentage of patients that have serum HDL within 3 standard deviations of the mean. (b) determine the percentage of patients that have serum HDL between 34 and 80.8. 56
Answer: (a) (1-1/9)*100%=88.9% (b) 57.4-34=23.4
80.8-57.4=23.4
23.4/11.7=2 two standard deviations, so the percentage of patients that have serum HDL between two stand deviations is at least (1-1/4)*100%=75% 57