CABT Math 8 measures of central tendency and dispersion

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Measures of Central Tendency and DispersionA Mathematics 8 Lecture

What the…?!

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

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


Branches of Statistics

DATA is any quantitative or qualitative information.


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)

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

The Measures of Central Tendency

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

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


The Mean

xx

n

Example 1


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

Example 2


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

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.


The Weighted Mean

wxx

w


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.


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

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


The Median

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


The Median


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

Example 5


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.

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)


The Mode


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


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

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.


The Mode

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


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



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

The Measures of Dispersion

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 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

Example 10


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


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


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


Mean DeviationExample 11Find the mean deviation of

3, 6, 6, 7, 8, 11, 15, 16

Solution

STEP 1: Find the mean:72

98

x


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



STEP 3: Add all the differences.


3 6

6 3

6 3

7 2

8 1

11 2

15 6

16 7

sum 30


3, 6, 6, 7, 8, 11, 15, 16



STEP 4: Divide the result by the number of data to get the MD:


3 6

6 3

6 3

7 2

8 1

11 2

15 6

16 7

sum 30


3, 6, 6, 7, 8, 11, 15, 16

303.75

8MD


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?


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


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


VarianceThe variance is simply the square of the standard deviation, or 2

2

2:x x

Variancen


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


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


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


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


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


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


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?


Standard Deviation

Remark:For both cases, the variance is simply the square of the standard deviation. The value is 2 74

Woohoo…



Find the standard deviation and variance of the following data set:

4, 8, 12, 15, 3, 2, 6, 9, 8, 7

Summing it

up!


That’s all for

today! Thank you!

Education

CABT Math 8 measures of central tendency and dispersion