Tech Math 2 Notes
Sections 22.1, 22,2, and 22.3
Page 1 of 6
Section 22.1: Frequency Distributions
Big Idea: The first step in analyzing data using statistics is to graph the data.
Section 22.2: Measures of Central Tendency
Big Idea: The second step in analyzing data using statistics is to find a single number that represents the most
probable value of all the data.
Section 22.3: A Measure of Spread: Standard Deviation
Big Idea: The third step in analyzing data using statistics is to find another single number that represents how
spread out the data is.
Using Statistics to Summarize Data
The term statistics refers to a collection of mathematical techniques for summarizing large amounts data.
In this section, we will learn how to make a histogram and calculate some examples of two kinds of statistics:
“measures of center” and “measures of spread.” Measures of center provide a single number that is meant to
represent all the data, like an average. Measures of spread provide a single number that is meant to characterize
how spread out the data is, like a range of the data. The measures of center we will calculate are the mean
(average), median, and mode, while the measure of spread we will calculate is the standard deviation. Two
other commonly used measures of spread that we will not calculate are the range and the inter-quartile range.
Steps for Statistically Analyzing Data:
1. Arrange the data in a table in ascending order.
a. Make a table with columns for the Datum #, Datum, Frequency, Deviation from Average, and
Square of Deviation.
b. Number each piece of data as you enter it in your table in ascending order.
c. Note the frequency of each datum as you enter it.
2. Make a histogram of the data.
a. A histogram is a bar chart showing the frequency for each different datum.
b. The horizontal axis should have a category for each datum.
c. The vertical axis is for the frequency.
d. Note: the curve that results from connecting the center of each bar of the histogram is called the
“distribution function.” The distribution function is used by statisticians to perform more
advanced statistics. If you had an infinite amount of “truly random” data, then the distribution
function would take the shape of a “bell-shaped curve” (or “normal curve”). The mean, median,
and mode are all the same for a normal curve, so the extent to which your data have different
values for those statistics is an indication of how different the data’s distribution function is from
a normal curve.
3. Make other graphs of the data, if requested
a. A frequency polygon is like a histogram except that you connect the number of data in each class
with line segments.
b. A cumulative frequency graph shows how the frequencies of each graph add up, and is also
sometimes called an ogive.
Tech Math 2 Notes
Sections 22.1, 22,2, and 22.3
Page 2 of 6
4. Calculate Measures of Center.
a. Mode – The mode is the most frequently occurring data value.
b. Median – The median is the data value such that one-half the data is less than or equal to the
median, and one-half the data is greater than or equal to the median. If there is an even number
of data, the median is the average of the two middle data values.
c. Mean (Average) – The mean is computed by adding up all the data, then dividing that sum by
the number of data. When you get comfortable calculating the mean, you can do it more quickly
by using the frequencies of the data. There are two different symbols for the mean:
i. Symbol for population mean (when you make a measurement on every member of the
population): .
 xf .
ii. Symbol for the (plain old) mean: x 
f
5. Calculate Measures of Spread.
a. Range – The range is the difference between the greatest and smallest data value.
b. Inter-Quartile Range – The inter-quartile range is the difference between the first quartile Q1
and the third quartile Q3.
i. First Quartile (Q1) – The first quartile is the data value such that one-fourth of the data
is less than or equal to the first quartile, and three-fourths of the data is greater than or
equal to the first quartile.
ii. Third Quartile (Q3) – The third quartile is the data value such that three-fourths of the
data is less than or equal to the third quartile, and one-fourth of the data is greater than or
equal to the third quartile
c. Standard Deviation – The standard deviation is the square root of the average of the squares of
the deviations (whew!). Just as there were two symbols for the mean, there are two symbols for
the standard deviation. However, each symbol represents a slightly different calculation:
i. Standard Deviation Symbol and Calculation: s
1. Calculate the “deviation from the average” for each datum by subtracting the
mean from the value of each datum.
2. Square each of the deviations.
3. Add up all the squares.
4. Divide the sum by n – 1 (one less than the number of data).
5. Take the square root. This number is the standard deviation.
6. Formula: s 
 x  x
2
n 1
ii. Population Standard Deviation Symbol and Calculation: 
1. Calculate the “deviation from the average” for each datum by subtracting the
mean from the value of each datum.
2. Square each of the deviations.
3. Add up all the squares.
4. Divide the sum by n (the number of data).
5. Take the square root. This number is the population standard deviation.
Tech Math 2 Notes
Sections 22.1, 22,2, and 22.3
Page 3 of 6
Use this data on the heights of people in two countries to practice these statistical techniques:
Height of People in Country A,
measured in inches:
Height of People in Country B,
measured in inches:
73, 71, 72, 72, 66, 70,
74, 72, 69, 71, 74, 67,
68, 72, 73, 71, 69, 70,
71, 72, 71, 72, 70, 73, 73
71, 72, 68, 75, 70, 72,
71, 73, 72, 74, 72, 75,
70, 69, 71, 71, 69, 73,
71, 70, 72, 74, 73, 70,
73, 71
1. Arrange the data in a table in ascending order. Use the table on the next page. Count the frequency
of each datum.
2. Make a histogram of the data.
Tech Math 2 Notes
Sections 22.1, 22,2, and 22.3
3. Make other graphs of the data.
4. Calculate measures of center.
a. Mode =
b. Median =
c. Mean s =  =
Note that adding the scores is faster using the frequencies:
Page 4 of 6
Tech Math 2 Notes
Sections 22.1, 22,2, and 22.3
5. Calculate measures of spread.
a. Range =
b. Inter-Quartile Range =
i. First Quartile Q1 =
ii. Third Quartile Q3 =
c. Standard Deviation
1. Standard deviation s =
2. Population standard deviation  =
Page 5 of 6
Tech Math 2 Notes
Sections 22.1, 22,2, and 22.3
Data Table for Country A
Datum #
Datum Frequency
Deviation from Average
( = Datum - Average ) Square of Deviation
Sum:
Sum:
Average:
Standard Deviation:
Population Standard
Deviation:
Page 6 of 6