1. Section 2.1
Review and Preview

2. Important Characteristics of Data
Preview
Important Characteristics of Data
1. Center: A representative or average value that indicates where the middle of the data set is located.
2. Variation: A measure of the amount that the data values vary.
3. Distribution: The nature or shape of the spread of data over the range of values (such as bell-shaped, uniform, or skewed).
4. Outliers: Sample values that lie very far away from the vast majority of other sample values.
5. Time: Changing characteristics of the data over time.
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3. Frequency Distributions
Section 2.2
Frequency Distributions

4. Key Concept
When working with large data sets, it is often helpful to organize and summarize data by constructing a table called a frequency distribution, defined later. Because computer software and calculators can generate frequency distributions, the details of constructing them are not as important as what they tell us about data sets. It helps us understand the nature of the distribution of a data set.
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5. Frequency Distributions
Definitions
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6. Frequency Distribution (or Frequency Table)
shows how a data set is partitioned among all of several categories (or classes) by listing all of the categories along with the number of data values in each of the categories.
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7. Pulse Rates of Females and Males
Original Data
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8. Frequency Distribution Pulse Rates of Females
The frequency for a particular class is the number of original values that fall into that class.
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9. Lower Class Limits Lower Class Limits
the smallest numbers that can actually belong to different classes
Lower Class
Limits
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10. Upper Class Limits Upper Class Limits
the largest numbers that can actually belong to different classes
Upper Class
Limits
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11. Class Boundaries Class Boundaries
the numbers used to separate classes, but without the gaps created by class limits
59.5
69.5
79.5
89.5
99.5
109.5
119.5
129.5
Class
Boundaries
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12. Class Midpoints Class Midpoints
are the values in the middle of the classes and can be found by adding the lower class limit to the upper class limit and dividing the sum by two.
64.5
74.5
84.5
94.5
104.5
114.5
124.5
Class
Midpoints
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13. Class Width Class Width
is the difference between two consecutive lower class limits or two consecutive lower class boundaries
Class
Width 10
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14. Example 1:
Identify the class width, class midpoints, and class boundaries for the given frequency distribution.
Tar (mg) in
Nonfiltered
Cigarettes
Frequency
10 – 13 1
14 – 17
18 – 21 15
22 – 25 7
26 – 29 2
Class Width: 4
Class Midpoints: 11.5, 15.5, 19.5, 23.5, 27.5
Class Boundaries: 9.5, 13.5, 17.5, 21.5, 25.5, 29.5
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15. Reasons for Constructing Frequency Distributions
1. Large data sets can be summarized.
2. We can analyze the nature of data.
3. We have a basis for constructing important graphs.
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16. Constructing A Frequency Distribution
1. Determine the number of classes (should be between 5 and 20).
2. Calculate the class width (round up).
class width 
(maximum value) – (minimum value)
number of classes
3. Starting point: Choose the minimum data value or a convenient value below it as the first lower class limit.
Using the first lower class limit and class width, proceed to list the other lower class limits.
5. List the lower class limits in a vertical column and proceed to enter the upper class limits.
6. Take each individual data value and put a tally mark in the appropriate class. Add the tally marks to get the frequency.
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17. Example 2:
Using the pulse rates of males in the given table, construct a frequency distribution. Use 7 classes.
Male Pulse Rates
Frequency
high– low # of classes
Class width =
= 96 –
56,57,58,59,60,61
Is 6 data values 13 7 9 2 4 1
Now count how many pulse rates fit in each category!
So each category needs to hold SIX pulse rates, starting with 56.
62 – 67
= 5.6 = 6
The difference between the 2 lower class limits is also SIX!
68 – 73
74 – 79
Continue this pattern for all 7 classes.
80 – 85
86 – 91
92 – 97
The total of this column should be 40, since there were 40 pulse rates provided!
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18. Relative Frequency Distribution
includes the same class limits as a frequency distribution, but the frequency of a class is replaced with a relative frequencies (a proportion) or a percentage frequency ( a percent)
relative frequency =
class frequency
sum of all frequencies
percentage
frequency
class frequency
sum of all frequencies
 100% =
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19. Relative Frequency Distribution
 12/40  100 = 30%
 1/40  100 = 2.5%
Total Frequency = 40
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20. Example 3:
Using the frequency distribution from Example 2, create a relative frequency distribution.
Male Pulse Rates
Frequency
56 – 61 13
62 – 67 7
68 – 73 9
74 – 79 2
80 – 85 4
86 – 91
92 – 97 1
Male Pulse Rates
Relative Frequency
56 – 61
62 – 67
68 – 73
74 – 79
80 – 85
86 – 91
92 – 97
32.5%
(13/40)
17.5%
(7/40)
22.5% 5%
10%
10%
2.5% 40
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21. Cumulative Frequency Distribution Cumulative Frequencies
includes the sum of the frequencies for that class and
all of the classes before it.
Cumulative Frequencies
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22. Frequency Table Comparison
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23. Cumulative Frequency Distribution
Example 4:
Using the frequency distribution from Example 2, create a cumulative frequency distribution.
Male Pulse Rates
Cumulative Frequency Distribution
Less than 62
Less than 68
Less than 74
Less than 80
Less than 86
Less than 92
Less than 98
Male Pulse Rates
Frequency
56 – 61 13
62 – 67 7
68 – 73 9
74 – 79 2
80 – 85 4
86 – 91
92 – 97 1 13 20
(13 + 7) 29
( ) 31 35 39 40
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24. Critical Thinking Interpreting Frequency Distributions
In later chapters, there will be frequent reference to data with a normal distribution. One key characteristic of a normal distribution is that it has a “bell” shape.
The frequencies start low, then increase to one or two high frequencies, then decrease to a low frequency.
The distribution is approximately symmetric, with frequencies preceding the maximum being roughly a mirror image of those that follow the maximum.
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25. Yes, the frequency starts small,
Example 5:
Using a strict interpretation of the relevant criteria on the previous slide, does the frequency distribution given below appear to have a normal distribution? Does the distribution appear to be normal if the criteria are interpreted very loosley?
Tar (mg) in
Nonfiltered
Cigarettes
Frequency
10 – 13 1
14 – 17
18 – 21 15
22 – 25 7
26 – 29 2
Yes, the frequency starts small,
Increases to a maximum,
Then decreases again.
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26. Weight (grams) of Pennies
Gaps
The presence of gaps can show that we have data from two or more different populations. However, the converse is not true, because data from different populations do not necessarily result in gaps.
Weight (grams) of Pennies
Frequency
2.40 – 2.49 18
2.50 – 2.59 19
2.60 – 2.69
2.70 – 2.79
2.80 – 2.89
2.90 – 2.99 2
3.00 – 3.09 25
3.10 – 3.19 8
Pennies were
once made copper,
but are now mostly
made out of zinc.
The gap could represent
the difference between
the material used.
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