cabt math 8 measures of central tendency and dispersion
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The session shall begin shortly…
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Measures of Central Tendency and DispersionA Mathematics 8 Lecture
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What the…?!
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What is Statistics?1. The science that deals with the collection,
organization, presentation, analysis, and interpretation of numerical data to obtain useful and meaningful information
2. A collection of quantitative data pertaining to a subject or group. Examples are blood pressure statistics etc.
A Brief Introduction to Statistics
What is Statistics
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Two branches of statistics:1. Descriptive Statistics:
Describes the characteristics of a product or process using information collected on it.
2. Inferential Statistics (Inductive):Draws conclusions on unknown process
parameters based on information contained in a sample.
Uses probability
A Brief Introduction to Statistics
Branches of Statistics
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DATA is any quantitative or qualitative information.
A Brief Introduction to Statistics
Data
Types of Data:
1. Quantitative – numerical information obtained from counting or measuring (e.g. age, qtr. exam scores, height)
2. Qualitative – descriptive attributes that cannot be subjected to mathematical operations (e.g. gender, religion, citizenship)
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Measures of Central Tendency and Dispersion
Statistics use numerical values used to summarize and compare sets of data.Measure of Central Tendency:
number used to represent the center or middle set of a set of data
Measure of Dispersion (or Variability): refers to the spread of values about the mean.
(i.e., how spread out the values are with respect to the mean)
The Measures of Central Tendency and Dispersion
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The Measures of Central Tendency
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The Measure of Central Tendency: 1. Mean - the (arithmetic) average (or
the sum of the quantities divided by the number of quantities)
2. Median – the middle value of a set of ordered data
3. Mode – number in a data set that occurs most frequently
Measures of Central Tendency and Dispersion
Measures of Central Tendency
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It’s known as the typical “average.” It is the most common measure of central
tendency. Symbolized as:
◦ for the mean of a sample◦ μ (Greek letter mu) for the mean of a
population• It’s equal to the sum of the quantities in
the data set divided by the number of quantities
x
Measures of Central Tendency and Dispersion
The Mean
xx
n
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Example 1
Measures of Central Tendency and Dispersion
The Mean
Find the mean of the numbers in the following data sets:
3 5 10 4 3 255
5 5x
b. 85, 87, 89, 90, 91, 98
a. 3, 5, 10, 4, 3
54090
6x
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Example 2
Measures of Central Tendency and Dispersion
The Mean
The table on the right shows the age of 13 applicants for a job in a factory in EPZA. What is the average age of the applicants?(Adapted from DOLE-BLES i-Learnstat module on Measures of Central Tendency)Solution:
31824.5
13x
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It is a mean where some values contribute more than others.
Each quantity is assigned a corresponding WEIGHT
(e.g. frequency or number, units, per cent) The weighted mean is equal to the sum
of the products of the quantities (x) and their corresponding weights (w), divided by the sum of the weights.
Measures of Central Tendency and Dispersion
The Weighted Mean
wxx
w
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Measures of Central Tendency and Dispersion
The Weighted MeanExample 3
SCORE NO. OF STUDENTS
5 8
4 6
3 3
2 2
1 1
The table shows the scores of 20 students in a 5-item Math IV seatwork.
Find the average score of the class.
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Measures of Central Tendency and Dispersion
The Weighted MeanExample 3 SolutionSCORE NO. OF
STUDENTS
5 8
4 6
3 3
2 2
1 1
Multiply the scores by the number of students, then find the sum. Finally, divide by the total number of students
PRODUCT
40
24
9
4
1
sums 20 7878
3.920
x
The average score is
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Used to find the middle value (center) of a distribution.
Used when one must determine whether the data values fall into either the upper 50% or lower 50% of a distribution.
Used when one needs to report the typical value of a data set, ignoring the outliers (few extreme values in a data set).◦ Example: median salary, median home prices in a market
Measures of Central Tendency and Dispersion
The Median
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How to find the median: Order the data in increasing order. If the number of data is ODD, the median
is the middle number.If n is odd, the middle number in n observations is the (n + 1)/2 th observation
If the number of data is EVEN, the median is the mean of the two middle numbers.
If n is even the middle number in n observations is the average of the (n/2)th and the (n/2+1)th observation
Measures of Central Tendency and Dispersion
The Median
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Measures of Central Tendency and Dispersion
The MedianExample 4Find the median of each set of data.a. 1, 2, 2, 3, 3, 4, 4, 5, 5b. 1, 2, 2, 3, 3, 4, 4, 4, 5, 5Answersa. Me = 3 (the 5th number)
b. The average of 5th and 6th numbers: 3 4
3.52
Me
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Example 5
Measures of Central Tendency and Dispersion
The Median
Solution:
Find the median of the following:3, 5, 6, 10, 9, 8, 7, 8, 9, 10, 7, 2, 5, 7
Arrange from lowest to highest:2, 3, 5, 5, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10
The median is 7.
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It is the number that appears most frequently in a set of data.
It is used when the most typical (common) value is desired.
It is not always unique. A distribution can have no mode, one mode, or more than one mode. When there are two or more modes, we say the distribution is multimodal.
(for two modes, we say that the distribution is bimodal)
Measures of Central Tendency and Dispersion
The Mode
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Measures of Central Tendency and Dispersion
The ModeExample 6
SCORE NO. OF STUDENTS
5 6
4 7
3 4
2 2
1 1
The table shows the scores of 20 students in a 5-item AP quiz.
What is the modal score?Answer: 4
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Measures of Central Tendency and Dispersion
The ModeExample 7
Find the mode of each set of data.
a. 1, 2, 2, 3, 3, 4, 4, 4, 5, 5
b. 1, 2, 2, 3, 3, 3,4, 4, 4, 5, 5
c. 1, 2, 3, 4, 5
Mo = 4
Mo = 3 and 4
No mode
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Example 8
Find the mode of the following:3, 5, 6, 10, 9, 8, 7, 8, 9, 10, 7, 2, 5, 7
Solution:Arrange from lowest to highest:2, 3, 5, 5, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10
The mode is 7.
Measures of Central Tendency and Dispersion
The Mode
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The data set above gives the waiting times (in minutes) of 10 students waiting for a bus. Find the mean,
median, and mode of the data set.
4, 8, 12, 15, 3, 2, 6, 9, 8, 7
Check your understanding
Measures of Central Tendency and Dispersion
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The data set above gives the waiting times (in minutes) of 10 students waiting for a bus. Find the
mean, median, and mode of the data set.
4, 8, 12, 15, 3, 2, 6, 9, 8, 7
Check your understanding
Measures of Central Tendency and Dispersion
Solution
74: 7.4 min
10Mean x
7 8: 7.5min
2Median Me
: 8minMode Mo
Arrange the data first in increasing order:2, 3, 4, 6, 7, 8, 8, 9, 12, 15
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The Measures of Dispersion
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The Measure of Dispersion or Variability1. Range – the difference of the largest
and smallest value2. Mean Absolute Deviation – the average
of the positive differences from the mean
3. Standard deviation – involves the average of the squared differences from the mean.
(related: variance)
Measures of Central Tendency and Dispersion
Measures of Dispersion
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Measures of Central Tendency and Dispersion
RangeSimply the difference between the
largest and smallest values in a set of data
Useful for analysis of fluctuations and for ordinal data
Is considered primitive as it considers only the extreme values which may not be useful indicators of the bulk of the population.
The formula is:Range = largest observation - smallest
observation
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Example 10
Measures of Central Tendency and Dispersion
Find the range of the following data sets: 10 3 7range
b. 85, 87, 89, 90, 91, 98
a. 3, 5, 10, 4, 3
98 85 13range
Range
c. 3, 5, 6, 10, 9, 8, 7, 8, 9, 10, 7, 2, 5, 7 10 2 8range
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Measures of Central Tendency and Dispersion
Mean DeviationIt measures the ‘average’ distance of
each observation away from the mean of the data
Gives an equal weight to each observation
Generally more sensitive than the range, since a change in any value will affect it
The formula is
where x is a quantity in the set, is the mean, and n is the number of data.
x xMD
n
x
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Measures of Central Tendency and Dispersion
Mean Deviation
To find the mean deviation:
1. Compute the mean.2. Get all the POSITIVE difference of
each number and the mean. (It’s the same as getting the absolute value of each difference)
3. Add all the results in step 2.4. Divide by the number of data.
x xMD
n
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Measures of Central Tendency and Dispersion
Mean DeviationExample 11Find the mean deviation of
3, 6, 6, 7, 8, 11, 15, 16
Solution
STEP 1: Find the mean:72
98
x
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Measures of Central Tendency and Dispersion
Mean DeviationExample 11
Find the mean deviation of
3, 6, 6, 7, 8, 11, 15, 16
STEP 2: Find the POSITIVE difference of each number and the mean (9).
VALUE POSITIVE DIFFERENCE
3 6
6 3
6 3
7 2
8 1
11 2
15 6
16 7
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Measures of Central Tendency and Dispersion
Mean DeviationExample 11
STEP 3: Add all the differences.
VALUE POSITIVE DIFFERENCE
3 6
6 3
6 3
7 2
8 1
11 2
15 6
16 7
sum 30
Find the mean deviation of
3, 6, 6, 7, 8, 11, 15, 16
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Measures of Central Tendency and Dispersion
Mean DeviationExample 11
STEP 4: Divide the result by the number of data to get the MD:
VALUE POSITIVE DIFFERENCE
3 6
6 3
6 3
7 2
8 1
11 2
15 6
16 7
sum 30
Find the mean deviation of
3, 6, 6, 7, 8, 11, 15, 16
303.75
8MD
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Measures of Central Tendency and Dispersion
Mean Deviation
It means that the quantities have an average difference of 3.75 from the mean (plus or minus).
What does the answer in the previous example mean?
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Measures of Central Tendency and Dispersion
Standard DeviationMeasures the variation of
observations from the meanThe most common measure of
dispersionTakes into account every
observationMeasures the ‘average deviation’ of
observations from the meanWorks with squares of residuals,
not absolute values—easier to use in further calculations
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Measures of Central Tendency and Dispersion
Standard DeviationThe formula for the standard
deviation is 2x x
n
where x is a quantity in the set, is the mean, and n is the number of data.
x
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Measures of Central Tendency and Dispersion
VarianceThe variance is simply the square of the standard deviation, or 2
2
2:x x
Variancen
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Measures of Central Tendency and Dispersion
Standard Deviation
To find the standard deviation:1. Compute the mean.2. Get the difference of each number and
the mean. 3. Square each difference4. Add all the results in step 3.5. Divide by the number of data.6. Get the square root.Note: If the VARIANCE is to be computed, skip the last step.
2x x
n
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Measures of Central Tendency and Dispersion
Standard DeviationPopulation versus Sample Standard DeviationThe standard deviation used here
is called the POPULATION standard deviation.
For very large populations, the SAMPLE standard deviation (s) is used. Its formula is 2
1
x xs
n
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Measures of Central Tendency and Dispersion
Standard DeviationAlternative Formula for the Standard DeviationAnother formula for standard
deviation uses only the sum of the data as well the sum of the squares of the data. This is
22n x x
n
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Measures of Central Tendency and Dispersion
Standard DeviationTo find the standard deviation using the alternative formula:1. Compute the squares of the data.2. Get the sum of the data and the sum of
the squares of the data.3. Multiply the sum of the squares by the
number of data, then subtract to the square of the sum of the data.
4. Get the square root of the result in step 3.
5. Divide the result by the number of data.
22n x x
n
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Measures of Central Tendency and Dispersion
Example 12Find the standard deviation of
3, 6, 6, 7, 8, 11, 15, 16using the given and the alternative formulas.Solution
Before using the formulas, it’s better to tabulate all results.
Standard Deviation
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Measures of Central Tendency and Dispersion
x x – x (x – x)2
3 –6 36
6 –3 9
6 –3 9
7 –2 4
8 –1 1
11 2 4
15 6 36
16 7 49
Standard Deviation
sum 148
Using the given formula
2x x
n
1488
4.3
2x x
n
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Measures of Central Tendency and Dispersion
x x2
3 9
6 36
6 36
7 49
8 64
11 121
15 225
16 256
Standard Deviation
sum 72 796
Using the alternative formula 22n x x
n
22n x x
n
28 796 72
8
1,1848
4.3
Ano ang
pipiliin mo?
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Measures of Central Tendency and Dispersion
Standard Deviation
Remark:For both cases, the variance is simply the square of the standard deviation. The value is 2 74
Woohoo…
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Check your understanding
Measures of Central Tendency and Dispersion
Find the standard deviation and variance of the following data set:
4, 8, 12, 15, 3, 2, 6, 9, 8, 7
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Summing it
up!
Measures of Central Tendency and Dispersion
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That’s all for
today! Thank you!