1) The document discusses various measures used to analyze grouped data, including mean, median, mode, range, and standard deviation. 2) It provides examples of calculating these measures from frequency distribution tables showing heights of students and exam marks of math students. 3) The document also discusses histograms and how they can visually display the distribution of data in grouped intervals.

1. Looking For Patterns in Grouped Data
Script Ohio by flickr
user Junior Sam

2. Looking For Patterns in Grouped Data
Script Ohio by flickr
user Junior Sam

3. Measures of Dispersion (Variability)
determine how quot;spread outquot; or variedquot; a set of data is.
Standard Deviation (σ): What's the difference between quot;σquot; and quot;squot;?
The symbol for standard deviation of a population or large
sample is quot;σquot; (sometimes written as quot;σx quot;), and the symbol for
standard deviation of a sample is 's'. A large sample is defined
as a sample with 30 or more data items. In this course, we will
use only quot;σquot; (sigma), which represents the standard deviation
of the population.
Let's take a look at a visual explanation of why we
calculate the standard deviation this way ...
http://www.seeingstatistics.com/

4. Working with Grouped Data
A frequency distribution table shows the number of elements of
data (frequency) at each measure. Sometimes the measures need
to be grouped, especially if the measures are continuous.
Example: The table below is a frequency distribution table that
shows the heights of 100 Senior 4 students. The students are
grouped into suitable height groups in 7 cm. intervals.
height interval interval mean # of students
153.5 to 160.5 157 5
160.5 to 167.5 164 16
167.5 to 174.5 171 43
174.5 to 181.5 178 27
181.5 to 188.5 185 9
Total 100
Determine the mean, median, mode, range, and standard
deviation for the student heights.
mean = 172.33 median = 171.00 mode = 171.00
range = 28 standard deviation = 6.837

5. height interval interval mean # of students
153.5 to 160.5 157 5
160.5 to 167.5 164 16
167.5 to 174.5 171 43
174.5 to 181.5 178 27
181.5 to 188.5 185 9
Total 100
Determine the mean, median, mode, range, and standard
deviation for the student heights.
mean = 172.33 median = 171.00 mode = 171.00
range = 28 standard deviation = 6.837

6. Working with Grouped Data
A probability distribution table shows the percent of elements of
data (probability) of each measure. Sometimes the measures need
to be grouped, especially if the measures are continuous.
Example: The table below is a frequency distribution table that
shows the heights of 100 Senior 4 students. The students are
grouped into suitable height groups in 7 cm. intervals.
height interval interval mean % of students
153.5 to 160.5 157 0.05
160.5 to 167.5 164 0.16
167.5 to 174.5 171 0.43
174.5 to 181.5 178 0.27
181.5 to 188.5 185 0.09
Total 1

7. Grouped Data and Histograms
A histogram is a bar graph that shows equal intervals of a
measured or counted quantity on the horizontal axis, and the
frequencies associated with these intervals on the vertical axis.
Drawing a histogram is useful because it shows the distribution of the
heights of the students.
A histogram is known as a Frequency Distribution Graph
when the data is obtained from a frequency distribution.
A histogram is known as a Probability Distribution Graph
when the data is obtained from a probability distribution.
Learn more about constructing a Histogram.
Click: Contents > 2. Seeing Data > 2.3 Histogram
http://www.seeingstatistics.com/

8. Let's apply what we've learned ...
mark interval mark # of students
The frequency distribution table 29 to 37 33 1
at right shows the midterm marks 38 to 46 42 4
of 85 Grade 12 math students. 47 to 55 51 12
The first column shows the mark 56 to 64 60 18
65 to 73 69 24
interval, the second column the
74 to 82 78 16
average mark within each mark 83 to 91 87 7
interval, and the third column the 92 to 100 96 3
number of students at each mark. Total 85
This is the
(a) Calculate the mean and std. dev. to two decimal places. correct way
Mean = 66.99 σ = 13.27 to do this.
(b) Calculate the number of students that have marks within one
std. dev. of the mean.
58 or 70
(c) What percent of students have marks within one std. dev. of
the mean?
68% or 82%

9. HOMEWORK
An experiment was performed to determine the approximate mass of
a penny. Three hundred pennies were weighed, and the weights
were recorded in the frequency distribution table shown below.
Mass (grams) 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4
Frequency 2 4 34 71 94 74 17 4
Determine the mean, median, mode, range, and standard deviation
of the data. Create a histogram that shows the frequencies of
different masses of this set of pennies.

10. HOMEWORK
No. of Videos
Four hundred people were No. of Persons
Returned
surveyed to find how many videos 1 28
they had rented during the last 2 102
month. Determine the mean and 3 160
median of the frequency 4 70
distribution shown below, and draw 5 25
6 13
a probability distribution
7 0
histogram. Also, determine the 8 2
mode by inspecting the frequency
distribution and the histogram.

11. HOMEWORK weight interval mean interval # of infants
The table shows the
3.5 to 4.5 4 4
weights (in pounds) of 125 4.5 to 5.5 5 11
newborn infants. The first 5.5 to 6.5 6 19
column shows the weight 6.5 to 7.5 7 33
7.5 to 8.5 8 29
interval, the second column
8.5 to 9.5 9 17
the average weight within 9.5 to 10.5 10 8
each weight interval, and 10.5 to 11.5 11 4
the third column the TOTAL 125
number of newborn infants
at each weight.
(a) Calculate the mean weight and standard deviation.
(b) Calculate the weight of an infant at one standard deviation
below the mean weight, and one standard deviation above the
mean.
(c) Determine the number of infants whose weights are within
one standard deviation of the mean weight.
(d) What percent of the infants have weights that are within one
standard deviation of the mean weight?