Chapter 3 Numerically Summarizing Data
Chapter 3.1Measures of Central Tendency
Objective A: Mean, Median and Mode
Three measures of central of tendency: the mean, the median, and the mode.
A1. Mean
The mean of a variable is the sum of all data values divided by the number of observations.
Population mean: where is each data value and is the population size (the number of
observations in the population).
Sample mean: where is each data value and n in the sample size (the number of
observations in the sample).
Example 1: Population: 12 16 23 17 32 27 14 16
Compute the population mean and sample mean from a simple random sample of size 4.
Does the sample mean equal to the population mean? Does the population mean or sample
mean stay the same? Explain.
(a) Population mean: (Round the mean to one more decimal place than that in the raw data)
(b) Sample mean:
From a lottery method, 23 16 14 17 were selected.
(c) Does the sample mean equal to the population mean?
(d) Does the population mean or sample mean stay the same? Explain.
A2. Median
The median, M, is the value that lies in the middle of the data when arranged in ascendingorder.
Example 1: Find the median of the data given below.
4 12 32 24 9 18 28 10 36
Example 2: Find the median of the data given below.
$35.34 $42.09 $38.72 $43.28 $39.45 $49.36 $30.15 $40.88
A3. Mode
Mode is the most frequent observation in the data set.
Example 1: Find the mode of the data given below.
76 60 81 72 60 80 68 73 80 67
Example 2: Find the mode of the data given below.
A C D C B C A B B F B W F D B W D A D C D
Example 3: The following data represent the G.P.A. of 12 students.
2.56 3.21 3.88 2.44 1.96 2.85 2.32 3.38 1.86 3.04 2.75 2.23
Find the mean, median, and mode G.P.A.
Objective B:Relation Between the Mean, Medianand DistributionShape
- The mean is sensitive to extreme data. For continuous data, if the distribution shape is
a bell-shaped curve, the mean is a better measure of central tendency because it
includesall data values in a data set.
- The median is resistant to extreme data. For continuous data, if the distribution shape is
skewed to the right or left, the median is a better measure of central tendency.
- The mode is used to represent the measure of central tendency for qualitative data.
Definition: 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.
Mean or Median versus Skewness
Chapter 3.2Measures of Dispersion
Measurement of dispersion is a numerical measure that can quantify the spread of data.
In this section, the three numerical measures of dispersion that we will discuss are the range, variance, and standard deviation. In the later section, we will discuss another measure of dispersion called interquartile range (IQR).
Objective A:Range, Variance and Standard Deviation
A1.Range
Range == largest data value - smallest data value
The range is not resistant because it is affected by extreme values in the data set.
A2. Variance and Standard Deviation
Variance is based on the deviation about the mean. Since the sum of deviation about the mean is zero, we
cannot use the average deviation about the mean as a measure of spread.
We use the average squared deviation instead.
The population variance, , of a variable is the sum of the squared deviations about the population
mean,, divided by the number of observations in the population, .
Definition Formula
Computational Formula
The sample variance, , of a variable is the sum of the squared deviations about the sample mean, ,
divided by the number of observations in the sample minus 1,.
Definition Formula
Computational Formula
In order to use the sample variance to obtain an unbiased estimate of the population variance, we divide
the sum of the squared deviations about the sample mean by .We call thedegree of freedom
because the first observations have freedomto be whatever value they wish, but the nth value has
no freedom in order to force to be zero.
The population standard deviation, , is the square root of the population variance or
The sample standard deviation, , is the square root of the sample variance or
To avoid round-off error, never use the rounded value of the variance to compute the standard deviation.
Keep a few more decimal places for an intermediate step calculation.
Example 1: Use the definition formula to find the population variance and standarddeviation.
Population: 4, 10, 12, 13, 21
Example 2: Use the definition formula to find the sample variance and standard deviation.
Sample: 83, 65, 91, 84
Example 3: Use StatCrunch to find the sample variance and standard deviation.
Sample: 83, 65, 91, 84 (same data set as Example 2)
Step 1:
Click StatCrunch navigation button under the Course Home page
Click StatCrunch website Click Open StatCrunch
Input the raw data in var 1 column Click Stat Click Summary Stats Columns
Step 2:
Click var1 under Select column(s): Under Statistics:, choose Variance and Std. dev. (click them while holding Ctrl key on the keyboard) Click Compute!
Variance and standard deviation are computed.
For more detailed instructions, please download “Q3.2.20 “ by clicking the StatCrunch Handout navigation button of the course homepage.
Note : For a small data set, students are expected to calculate the standard deviation by hand.
Objective C : Empirical Rule
The figure below illustrates the Empirical Rule
Example 1: SAT Math scores have a bell-shaped distribution with a mean of 515 and a standard
deviation of 114. (Source: College Board, 2007)
(a) What percentage of SAT scores is between 401 and 62
(b) What percentage of SAT scores is between 287 and 743?
(c) What percentage of SAT scores is less than 401 or greater than 629?
(d) What percentage of SAT scores is between 515 and 743?
(e) About 99.7% of SAT scores will be between what scores?
Chapter 3.4Measures of Position and Outliers
Measures of position determine the relative position of a certain data value within the entire set of data.
Objective A :-scores
The z-score represents the distance that a data value is from the mean in terms of the number
of standard deviations.
Population -score: Sample -score:
Example 1: The average 20- to 29-year-old man is 69.6 inches tall, with a standard deviation of
3.0 inches, while the average 20- to 29-year-old woman is 64.1 inches tall, with a standard deviation of 3.8 inches. Who is relativelytaller, a 67-inch man or 62-inch woman?
Objective B:Percentiles and Quartiles
B1. Percentiles
The th percentile, , of a set of data is a value such that percent of the observationsare less
than or equal to the value.
Example 1: Explain the meaning of the 5th percentile of the weight of males 36 months of age is
12.0 kg.
The most common percentiles are quartiles.
The first quartile, , is equivalent to .
The second quartile, , is equivalent to .
The third quartile, , is equivalent to .
Example 2: Determine the quartiles of the following data.
46 45 58 71 42 66 72 42 61 49 80
B2. Interquartile
The interquartile range, IQR, is the measure of dispersion that is based on quartiles. The range and
standard deviation are affected by extreme values. The IQR is resistant to extreme values.
Example 1: One variable that is measured by online homework systems is the amount of time a student
spends on homework for each section of the text. The following is a summary of the number
of minutes a student spends for each section of the text for the fall 2007 semester in a
College Algebra class at Joliet Junior College.
(a) Provide an interpretation of .
(b) Determine and interpret the interquartile range.
(c) Do you believe that the distribution of time spent doing homework isskewed or
symmetric? Why?
Objective C:Outliers
Extreme observations are called outliers; they may occur by error in the measurement or during data
entry or from errors in sampling.
Chapter 3.5 The Five-Number Summary and Boxplots
Objective A : The Five-Number Summary
Example 1: The number of chocolate chips in a randomly selected 21 name-brand cookies
were recorded. The results are shown below.
28 23 28 31 27 29 2419 26 23 21 25 22 23
21 23 33 28 33 21 30
Find the five-number Summary.
Objective B: Boxplots
The five-number summary can be used to construct a graph called the boxplot.
Objective C: Using a Boxplot to describe the shape of a distribution
Example 1: Use the side-by-side boxplots shown to answer the questions that follow.
(a) To the nearest integer, what is the median of variable?
(b) To the nearest integer, what is the first quartile of variable?
(c) Which variable has more dispersion? Why?
(d) Does the variable have any outliers? If so, what is the value ofthe outlier?
(e) Describe the shape of the variable. Support your position.
1