Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice...
Transcript of Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice...
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Measures of
Central
TendencyFor Ungrouped Data
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Recall:What are the three ways
on how we can define
“central tendency” or
“center of the
distribution”?
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We can also say:
(1) the point on
which a distribution
would be balance;
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We can also say:(2) the value whose
average absolute
deviation from all other
values is minimized;
and
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We can also say:(3) the value whose
average squared
difference from all the
other values is
minimized.
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Recall:
What are the
measures of central
tendency?
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Do you know?
The mean is the
point on which the
distribution would
balance.
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Do you know?The median is the
value that minimizes
the sum of absolute
deviations.
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Do you know?The mean is the
value that minimizes
the sum of squared
deviations.
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The
MeanFor Ungrouped Data
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First
TypeArithmetic Mean
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Arithmetic MeanIt is also called as “simple
mean” or “unweighted
mean”. It is the sum of a
collection of numbers
divided by the number of
numbers in the collection.
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FormulaArithmetic Mean:
𝒙 =𝚺𝒙
𝒏where: x = items/scores
n=number of items/scores
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Example:What is the mean age
of a group of children
whose ages are 9, 11, 7,
10, 9, 8, 8, 7, 12, 7 and
13?
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Second
TypeWeighted Mean
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Weighted MeanIt takes into
consideration the proper
weights assigned to the
observed values
according to importance.
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FormulaWeighted Mean:
𝒙 =𝚺𝒘𝒙
𝒘where: x=items/scores
w=weight of each item/score
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Example:A student took 3 exams in Math. He
finished the first exam in 45 minutes
and got a grade of 88; 60 minutes
on the second exam and got 92;
and 90 minutes on the third exam
and got 85. What was the student’s
mean score for the three exams?
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Third
TypeMean for Simple Frequency
Distribution
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FormulaMean for Simple Frequency
Distribution:
𝒙 =𝚺𝒇𝒙
𝒏Where: f=frequency; x=items/scores;
and n=number of items
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Example:x f 𝚺fx
8 2
2 3
6 4
7 3
5 5
6 2
9 1
n=20 𝚺fx=
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Example:x f 𝚺fx
8 2 16
2 3 6
6 4 24
7 3 21
5 5 25
6 2 12
9 1 9
n = 20 𝚺fx = 114
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Let’s
PracticeMeasures of Central Tendency
for Ungrouped Data
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Problem 1:A freshman college student for the
following final grades with the
number of units for each subject. (a)
Find the weighted mean grade and
(b) if all the subjects have uniform or
equal number of units, what would
be the mean grade?
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Table 1. A freshman college student
final grades with the number of units
for each subject.Subjects Grades Units
Math 1 2.50 3
English 1 2.00 3
Filipino 1 2.50 3
History 1 1.75 3
PE 1 1.50 2
Chemistry Lecture 3.00 2
Chemistry Laboratory 2.50 1
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Problem 2:Gottfried Wilhelm Leibniz answered
20 calculus problems. He spent 1 ½
hours for the first 6 problems; 45
minutes for 3 problems; and 3 hours
for 11 problems. What was the
average time he spent for each
problem?
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Recall:What are the
symbols we use for
each of measure of
central tendency?
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Something to think about…
If you are given a
dataset/frequency
distribution that is already
grouped, what will be the
most challenging part?
Why?
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Measures of
Central
TendencyFor Grouped Data
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The
MeanFor Grouped Data
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Long Method/Midpoint
MethodFormula:
𝒙 =𝚺𝒇𝒙
𝒏Where: f=frequency; x=class
marks; and n=number of samples
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Example:
Class Intervalsf
(frequency)
x
(class mark)
fx
(frequency x
class mark)
62-68 8
69-75 5
76-82 9
83-89 11
90-96 12
97-103 10
104-110 5
111-117 5
n=65
![Page 33: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/33.jpg)
Example:
Class Intervalsf
(frequency)
x
(class mark)
fx
(frequency x
class mark)
62-68 8 65 520
69-75 5 72 360
76-82 9 79 711
83-89 11 86 946
90-96 12 93 1,116
97-103 10 100 1,000
104-110 5 107 535
111-117 5 114 570
n=65 𝚺fx=5,758
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Short Method or
Assumed Mean Method
It is the class mark of the class
interval near the center of the
distribution or the class mark of
the interval with the highest
frequency.
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FormulaAssumed Mean
Method:
𝒙 = 𝒙𝟎 +𝚺𝒇𝒅
𝒏𝒘
![Page 36: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/36.jpg)
Where:𝒙𝟎 = 𝒂𝒔𝒔𝒖𝒎𝒆𝒅𝒎𝒆𝒂𝒏
𝒇 = 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚𝒏 = 𝒔𝒊𝒎𝒑𝒍𝒆 𝒔𝒊𝒛𝒆𝒅 = 𝒄𝒐𝒅𝒆𝒅 𝒗𝒂𝒍𝒖𝒆𝒘 = 𝒄𝒍𝒂𝒔𝒔 𝒘𝒊𝒅𝒕𝒉
![Page 37: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/37.jpg)
Example:Class
Intervals
f
(frequency)
x
(Class Mark)
fx
(frequency x
class mark)
fd
62-68 8
69-75 5
76-82 9
83-89 11
90-96 12
97-103 10
104-110 5
111-117 5
n=65
![Page 38: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/38.jpg)
Steps involved in
the calculation of
Mean by the
Short MethodFor Grouped Data
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Step 1:
Tabulate the scores
in a frequency
distribution.
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Example:Class
Intervals
f
(frequency)
x
(Class Mark)
fx
(frequency x
class mark)
fd
62-68 8
69-75 5
76-82 9
83-89 11
90-96 12
97-103 10
104-110 5
111-117 5
n=65
![Page 41: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/41.jpg)
Step 1:Take the midpoint of an interval
somewhere near the center of
the frequency distribution and,
if possible, the interval should
contain the largest frequency.
![Page 42: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/42.jpg)
Example:Class
Intervals
f
(frequency)
x
(Class Mark)
fx
(frequency x
class mark)
fd
62-68 8 65
69-75 5 72
76-82 9 79
83-89 11 86
90-96 12 93
97-103 10 100
104-110 5 107
111-117 5 114
n=65
![Page 43: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/43.jpg)
Note:
That will be our
assumed mean.
Therefore, 𝒙𝟎 = 𝟗𝟑.
![Page 44: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/44.jpg)
Step 3:Fill in the column for the deviations
from the assumed mean in units of
class intervals (d). Starting with 0 for the class interval having the 𝒙𝟎
going up assign positive values and
going down assign negative values
for d.
![Page 45: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/45.jpg)
Example:Class
Intervals
f
(frequency)
x
(Class Mark)
d
(coded
value)
fd
62-68 8 65 -4
69-75 5 72 -3
76-82 9 79 -2
83-89 11 86 -1
90-96 12 93 0
97-103 10 100 +1
104-110 5 107 +2
111-117 5 114 +3
n=65
![Page 46: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/46.jpg)
Step 4:The fd column is the product
of f and d. The values greater than the 𝒙𝟎 are
positive and the values less than 𝒙𝟎are negative.
![Page 47: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/47.jpg)
Example:Class
Intervals
f
(frequency)
x
(Class Mark)
d
(coded
value)
fd
62-68 8 65 -4 -32
69-75 5 72 -3 -15
76-82 9 79 -2 -18
83-89 11 86 -1 -11
90-96 12 93 0 0
97-103 10 100 +1 +10
104-110 5 107 +2 +10
111-117 5 114 +3 +15
n=65
![Page 48: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/48.jpg)
Step 5:
Get the sum of the 𝒇𝒅 values.
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Example:Class
Intervals
f
(frequency)
x
(Class Mark)
d
(coded
value)
fd
62-68 8 65 -4 -32
69-75 5 72 -3 -15
76-82 9 79 -2 -18
83-89 11 86 -1 -11
90-96 12 93 0 0
97-103 10 100 +1 +10
104-110 5 107 +2 +10
111-117 5 114 +3 +15
n=65 𝚺fd=-41
![Page 50: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/50.jpg)
Step 6:
Determine the
class width
(w).
![Page 51: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/51.jpg)
Therefore,
The class
width is 7.
![Page 52: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/52.jpg)
Step 7:
Substitute the
values obtained
to the formula.
![Page 53: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/53.jpg)
Therefore,
The𝒙 = 𝟖𝟖. 𝟔.
![Page 54: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/54.jpg)
Something to think about…
What can you say to the
mean we obtained using
the Midpoint Method and
Assumed Mean Method?
![Page 55: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/55.jpg)
The
MedianFor Grouped Data
![Page 56: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/56.jpg)
Formula:
𝒙 = 𝑳𝑳𝑹 +
𝒏𝟐− 𝒇 ≤
𝒇𝒘
![Page 57: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/57.jpg)
Where:
𝑳𝑳𝑹 = 𝒍𝒐𝒘𝒆𝒓 𝒓𝒆𝒂𝒍 𝒍𝒊𝒎𝒊𝒕𝒇 ≤= 𝒄𝒖𝒎𝒖𝒍𝒂𝒕𝒊𝒗𝒆 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚
𝒇 = 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚𝒏 = 𝒔𝒂𝒎𝒑𝒍𝒆
𝒘 = 𝒄𝒍𝒂𝒔𝒔 𝒔𝒊𝒛𝒆
![Page 58: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/58.jpg)
Example:Class Intervals
f
(frequency)
f≤
(cumulative less than
frequency)
62-68 8
69-75 5
76-82 9
83-89 11
90-96 12
97-103 10
104-110 5
111-117 5
n=65
![Page 59: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/59.jpg)
Steps Involved in
the Calculation
of MedianFor Grouped Data
![Page 60: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/60.jpg)
Step 1:
Record the
cumulative
frequencies.
![Page 61: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/61.jpg)
Example:Class Intervals
f
(frequency)
f≤
(cumulative less than
frequency)
62-68 8 8
69-75 5 13
76-82 9 22
83-89 11 33
90-96 12 45
97-103 10 55
104-110 5 60
111-117 5 65
n=65
![Page 62: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/62.jpg)
Step 2:
Determine 𝒏
𝟐.
![Page 63: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/63.jpg)
Therefore,
𝒏
𝟐= 𝟑𝟐. 𝟓.
![Page 64: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/64.jpg)
Step 3:
Identify the class
interval in which
the 32.5th case falls.
![Page 65: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/65.jpg)
Example:Class Intervals
f
(frequency)
f≤
(cumulative less than
frequency)
62-68 8 8
69-75 5 13
76-82 9 f≤=22
83-89 f=11 33
90-96 12 45
97-103 10 55
104-110 5 60
111-117 5 65
n=65
![Page 66: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/66.jpg)
Therefore,
it is the 4th class
interval with exact
limits 82.5-89.5. 𝑳𝑳𝑹is 82.5.
![Page 67: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/67.jpg)
Step 4:
Determine the
class width
(w).
![Page 68: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/68.jpg)
Therefore,
the class
width is 7.
![Page 69: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/69.jpg)
Step 5:
Substitute the
values obtained
to the formula.
![Page 70: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/70.jpg)
Answer:
The median
is 89.2
![Page 71: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/71.jpg)
The
ModeFor Grouped Data
![Page 72: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/72.jpg)
Formula:
𝒙 = 𝑳𝑳𝑹 +𝒅𝒖
𝒅𝒖 + 𝒅𝒍𝒘
![Page 73: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/73.jpg)
Where:𝑳𝑳𝑹 = 𝒍𝒐𝒘𝒆𝒓 𝒓𝒆𝒂𝒍 𝒍𝒊𝒎𝒊𝒕
𝒅𝒖 = difference between the highest
frequency and the frequency of the
interval below it
𝒅𝒍 = difference between the highest
frequency and the frequency of the
interval above it
𝒘 =class size
![Page 74: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/74.jpg)
Example:Class Intervals f
62-68 8
69-75 5
76-82 9
83-89 11
90-96 12
97-103 10
104-110 5
111-117 5
n=65
![Page 75: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/75.jpg)
Steps Involved in
the Calculation
of ModeFor Grouped Data
![Page 76: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/76.jpg)
Step 1:
Tabulate the scores
in a frequency
distribution.
![Page 77: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/77.jpg)
Example:Class Intervals f
62-68 8
69-75 5
76-82 9
83-89 11
90-96 12
97-103 10
104-110 5
111-117 5
n=65
![Page 78: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/78.jpg)
Step 1:
Determine the class
interval with the
highest frequency.
![Page 79: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/79.jpg)
Example:Class Intervals f
62-68 8
69-75 5
76-82 9
83-89 11
90-96 12
97-103 10
104-110 5
111-117 5
n=65
![Page 80: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/80.jpg)
Then,
the 𝑳𝑳𝑹 will be the
lower real limit of the
class interval with the
highest frequency.
![Page 81: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/81.jpg)
Thus,
𝑳𝑳𝑹 = 𝟖𝟗. 𝟓
![Page 82: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/82.jpg)
Step 2:Determine 𝒅𝒖. 𝒅𝒖 is the
difference between the
highest frequency and
the frequency of the
interval below it.
![Page 83: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/83.jpg)
Example:Class Intervals f
62-68 8
69-75 5
76-82 9
83-89 11
90-96 12
97-103 10
104-110 5
111-117 5
n=65
![Page 84: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/84.jpg)
Thus,
𝒅𝒖 = 𝟏𝟐 − 𝟏𝟏 = 𝟏
![Page 85: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/85.jpg)
Step 3:Determine 𝒅𝒍. 𝒅𝒍 is the
difference between the
highest frequency and
the frequency of the
interval above it.
![Page 86: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/86.jpg)
Example:Class Intervals f
62-68 8
69-75 5
76-82 9
83-89 11
90-96 12
97-103 10
104-110 5
111-117 5
n=65
![Page 87: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/87.jpg)
Thus,
𝒅𝒍 = 𝟏𝟐 − 𝟏𝟎 = 𝟐
![Page 88: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/88.jpg)
Step 4:
Determine the
class width.
![Page 89: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/89.jpg)
Thus,
the class
width is 7.
![Page 90: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/90.jpg)
Step 5:
Substitute the
values obtained
to the formula.
![Page 91: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/91.jpg)
Thus,
the mode is
91.8.
![Page 92: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/92.jpg)
Kinds of
ModeFor Grouped Data
![Page 93: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/93.jpg)
The True
ModeKinds of Mode
![Page 94: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/94.jpg)
Formula:
𝒙 = 𝟑𝑴𝒆𝒅𝒊𝒂𝒏 − 𝟐𝑴𝒆𝒂𝒏
![Page 95: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/95.jpg)
From the example:
using the true
mode formula,
the mode is 90.4.
![Page 96: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/96.jpg)
Crude
ModeKinds of Mode
![Page 97: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/97.jpg)
Crude Mode
It is the midpoint of
the class interval
with the highest
frequency.
![Page 98: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/98.jpg)
Formula:
𝒙 =𝒍𝒐𝒘𝒆𝒓 𝒍𝒊𝒎𝒊𝒕 + 𝒖𝒑𝒑𝒆𝒓 𝒍𝒊𝒎𝒊𝒕
𝟐
![Page 99: Measures of Central Tendencymigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter...Practice Measures of Central Tendency for Ungrouped Data Problem 1: A freshman college student](https://reader033.fdocuments.us/reader033/viewer/2022061005/60b32e40d16caa6b97798cc8/html5/thumbnails/99.jpg)
From the example:
using crude
mode method,
the mode is 93.