Matematika Bahasa Inggris

PREFACE

First of all, thanks to Allah SWT because of the help of Allah, writer finished writing the paper entitled “DATA” right in the calculated time.

The purpose in writing this paper is to fulfill the assignment that given by Mrs. NurinaAyuningtyas, S.Pd, M.Pd.as lecturer in semantics major.

In arranging this paper, the writer truly get lots challenges and obstructions but with help of many individuals, those obstructions could passed writer also realized there are still many mistakes in process of writing this paper.

Sidoarjo, 12 Oktober 2016

Writer

CAPTHER 1

INTRODUCTION

1.1              Background

                        Statistics is a set of methods that are used to collect, analyze, present, and interpret data. Statistical methods are used in a wide variety of occupations and help people identify, study, and solve many complex problems. In the business and economic world, these methods enable decision makers and managers to make informed and better decisions about uncertain situations.

                        Vast amounts of statistical information are available in today's global and economic environment because of continual improvements in computer technology. To compete successfully globally, managers and decision makers must be able to understand the information and use it effectively. Statistical data analysis provides hands on experience to promote the use of statistical thinking and techniques to apply in order to make educated decisions in the business world.

                        Decision making process under uncertainty is largely based on application of statistical data analysis for probabilistic risk assessment of your decision. Managers need to understand variation for two key reasons. First, so that they can lead others to apply statistical thinking in day to day activities and secondly, to apply the concept for the purpose of continuous improvement. This course will provide you with hands-on experience to promote the use of statistical thinking and techniques to apply them to make educated decisions whenever there is variation in business data. Therefore, it is a course in statistical thinking via a data-oriented approach.

                        There are two general views of teaching/learning statistics. For example,  Greater and Lesser Statistics. Greater statistics is everything related to learning from data, from the first planning or collection, to the last presentation or report. Lesser statistics is the body of statistical methodology. This is a Greater Statistics course.

1.2              Problem Identification

1.      What is the meaning of data and statistics?

2.      How to aplicate statistics in our activity?

1.3              Purpose

1.      To understanding the description of data and statistics.

2.      To identivication the application in our activities or jobs.

CHAPTER II

DISCUSSION

2.1. Data

A. What is Data?

Data is a collection of facts, such as numbers, words, measurements, observations or even just descriptions of things.

Data can be qualitative or quantitative.

·         Qualitative data is descriptive information (it describes something)

·         Quantitative data, is numerical information (numbers).

B. Discrete and Continuous Data

And Quantitative data can also be Discrete or Continuous:

Discrete and Continuous Data

Data can be Descriptive (like "high" or "fast") or Numerical (numbers).

And Numerical Data can be Discrete or Continuous:

Discrete data is counted, Continuous data is measured

Ø Discrete Data

Discrete Data can only take certain values.

Example: the number of students in a class (you can't have half a student).

Example: the results of rolling 2 dice:

can only have the values 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12

Ø   Continuous Data

Continuous Data can take any value (within a range)

Examples:

·         A person's height: could be any value (within the range of human heights), not just certain fixed heights,

·         Time in a race: you could even measure it to fractions of a second,

·         A dog's weight,

·         The length of a leaf,

·         Lots more!

·         Discrete data can only take certain values (like whole numbers)

·         Continuous data can take any value (within a range)

Put simply: Discrete data is counted, Continuous data is measured

Example: What do we know about Arrow the Dog?

Qualitative:

·         He is brown and black

·         He has long hair

·         He has lots of energy

Quantitative:

·Discrete:

o    He has 4 legs

o    He has 2 brothers

·Continuous:

o    He weighs 25,5 kg

o    He is 565 mm tall

More Examples

Qualitative:

·            Your friends' favorite holiday destination

·            The most common given names in your town

·            How people describe the smell of a new perfume

Quantitative:

·            Height (Continuous)

·            Weight (Continuous)

·            Petals on a flower (Discrete)

·            Customers in a shop (Discrete)

Collecting

Data can be collected in many ways. The simplest way is direct observation.

Example: you want to find how many cars pass by a certain point on a road in a 10-minute interval.

So: stand at that point on the road, and count the cars that pass by in that interval.

We collect data by doing a Survey.

Census or Sample

A Census is when we collect data for every member of the group (the whole "population").

A Sample is when we collect data just for selected members of the group.

Example: there are 120 people in your local football club.

You can ask everyone (all 120) what their age is. That is a census.

Or you could just choose the people that are there this afternoon. That is a sample.

A census is accurate, but hard to do. A sample is not as accurate, but may be good enough, and is a lot easier.

2.2.            How to Show Data

A.    Bar Graph

A Bar Graph (also called Bar Chart) is a graphical display of data using bars of different heights.

Imagine you just did a survey of your friends to find which kind of movie they liked best:

Table:Favorite Type of Movie
Comedy	Action	Romance	Drama	SciFi
4	5	6	1	4

We can show that on a bar graph like this:

It is a really good way to show relative sizes: we can see which types of movie are most liked, and which are least liked, at a glance.

We can use bar graphs to show the relative sizes of many things, such as what type of car people have, how many customers a shop has on different days and so on.

Example: Nicest Fruit

Fruit:	Apple	Orange	Banana	Kiwifruit	Blueberry	Grapes
People:	35	30	10	25	40	5

A survey of 145 people asked them "Which is the nicest fruit?":

And here is the bar graph:

That group of people think Blueberries are the nicest.

Bar Graphs can also be Horizontal, like this:

Example: Student Grades

In a recent test, this many students got these grades:

Grade:	A	B	C	D
Students:	4	12	10	2

And here is the bar graph:

Histograms vs Bar Graphs

Bar Graphs are good when your data is in categories (such as "Comedy", "Drama", etc).

But when you have continuous data(such as a person's height) then use a Histogram.

It is best to leave gaps between the bars of a Bar Graph, so it doesn't look like a Histogram.

Histograms

Histogram: a graphical display of data using bars of different heights.

It is similar to a  but a histogram groups numbers into ranges And you decide what ranges to use!

Example: Height of Orange Trees

You measure the height of every tree in the orchard in centimeters (cm)

The heights vary from 100 cm to 340 cm

You decide to put the results into groups of 50 cm:

·       The 100 to just below 150 cm range,

·       The 150 to just below 200 cm range,

·       etc...

So a tree that is 260 cm tall is added to the "250-300" range.

And here is the result: You can see (for example) that there are 30 trees from 150 cm to just below 200 cm tall

The horizontal axis is continuous like a number line:

Example: How much is that puppy growing?

Each month you measure how much weight your pup has gained and get these results:

0,5, 0,5, 0,3, −0,2, 1,6, 0, 0,1, 0,1, 0,6, 0,4

They vary from −0,2 (the pup lost weight that month) to 1,6

Put in order from lowest to highest weight gain:

−0,2, 0, 0,1, 0,1, 0,3, 0,4, 0,5, 0,5, 0,6, 1,6

You decide to put the results into groups of 0,5:

·       The −0,5 to just below 0 range,

·       The 0 to just below 0,5 range,

·       etc...

And here is the result: There are no values from 1 to just below 1,5, but we still show the space:

The range of each bar is also called the Class Interval

In the example above each class interval is 0,5 

Histograms are a great way to show results of continuous data, such as:

·         weight

·         height

·         how much time

·        

etc.

But when the data is in categories (such as Country or Favorite Movie), we should use a Bar Chart.

Frequency Histogram

A Frequency Histogram is a special histogram that uses vertical columns to show frequencies (how many times each score occurs):

Here I have added up how often 1 occurs (2 times), how often 2 occurs (5 times), etc, and shown them as a histogram.

B.     Cumulative Tables and Graphs

o    Cumulative

Cumulative means "how much so far".

Think of the word "accumulate" which means to gather together.

To have cumulative totals, just add up the values as you go.

Example: Jamie has earned this much in the last 6 months:

Month	Earned
March	$120
April	$50
May	$110
June	$100
July	$50
August	$20

To work out the cumulative totals, just add up as you go.

The first line is easy, the total earned so far is the same as Jamie earned that month:

Month	Earned	Cumulative
March	$120	$120

But for April, the total earned so far is $120 + $50 = $170 :

Month	Earned	Cumulative
March	$120	$120
April	$50	$170

And for May we continue to add up: $170 + $110 = $280

Month	Earned	Cumulative
March	$120	$120
April	$50	$170
May	$110	$280

Do you see how we add the previous month's cumulative total to this month's earnings?

Here is the calculation for the rest:

·         June is $280 + $100 = $380

·         July is $380 + $50 = $430

·         August is $430 + $20 = $450

And this is the result

Month	Earned	Cumulative
March	$120	$120
April	$50	$170
May	$110	$280
June	$100	$380
July	$50	$430
August	$20	$450

The last cumulative total should match the total of all earnings:

$450 is the last cumulative total ...
... it is also the total of all earnings:

$120+$50+$110+$100+$50+$20 = $450

So we got it right.

So that's how to do it, add up as you go down the list and you will have cumulative totals.

We could also call it a "Running Total"

Graphs

We can make cumulative graphs, too. Just plot each cumulative total:

Cumulative Bar Graph
Cumulative Line Graph

C.     Dot Plots

A Dot Plot is a graphical display of data using dots.

Example: Minutes To Eat Breakfast

A survey of "How long does it take you to eat breakfast?" has these results:

Minutes:	0	1	2	3	4	5	6	7	8	9	10	11	12
People:	6	2	3	5	2	5	0	0	2	3	7	4	1

Which means that 6 people take 0 minutes to eat breakfast (they probably had no breakfast!), 2 people say they only spend 1 minute having breakfast, etc.

And here is the dot plot:

Another version of the dot plot has just one dot for each data point like this:

Example: (continued)

This has the same data as above:

But notice that we need to have lines and numbers on the side so we can see what the dots mean.

Grouping

Example: Access to Electricity across the World

Some people don't have access to electricity (they live in remote or poorly served areas). A survey of many countries had these results:

Country	Access to Electricity (% of population)
Algeria	99,4
Angola	37,8
Argentina	97,2
Bahrain	99,4
Bangladesh	59,6
...	... etc

But hang on! How do we make a dot plot of that? There might be only one "59,6" and one "37,8", etc. Nearly all values will have just one dot.

The answer is to group the data (put it into "bins").

In this case let's try rounding every value to the nearest 10%:

Country	Access to Electricity (% of population, nearest 10%)
Algeria	100
Angola	40
Argentina	100
Bahrain	100
Bangladesh	60
...	... etc

Now we count how many of each 10% grouping and these are the results:

Access to Electricity (% of population, nearest 10%)	Number of Countries
10	5
20	6
30	12
40	5
50	4
60	5
70	6
80	10
90	15
100	34

So there were 5 countries where only 10% of the people had access to electricity, 6 countries where 20% of the people had access to electricity, etc

Here is the dot plot:

Percent of Population with Access to Electricity

And that is a good plot, it shows the data nicely.

D.    Frequency Distribution

Frequency

Frequency is how often something occurs.

Example: Sam played football on

·         Saturday Morning,

·         Saturday Afternoon

·         Thursday Afternoon

The frequency was 2 on Saturday, 1 on Thursday and 3 for the whole week.

Frequency Distribution

By counting frequencies we can make a Frequency Distribution table.

Example: Goals

Sam's team has scored the following numbers of goals in recent games:

2, 3, 1, 2, 1, 3, 2, 3, 4, 5, 4, 2, 2, 3

Sam put the numbers in order, then added up:

·         how often 1 occurs (2 times),

·         how often 2 occurs (5 times),

·         etc,

and wrote them down as a Frequency Distribution table.

From the table we can see interesting things such as

·       getting 2 goals happens most often

·       only once did they get 5 goals

This is the definition:

Frequency Distribution: values and their frequency (how often each value occurs).

Here is another example:

Example: Newspapers

These are the numbers of newspapers sold at a local shop over the last 10 days:

22, 20, 18, 23, 20, 25, 22, 20, 18, 20

Let us count how many of each number there is:

Papers Sold	Frequency
18	2
19	0
20	4
21	0
22	2
23	1
24	0
25	1

It is also possible to group the values. Here they are grouped in 5s:

Papers Sold	Frequency
15-19	2
20-24	7
25-29	1

(Learn more about Grouped Frequency Distributions)

E.     Line Graphs

Line Graph: a graph that shows information that is connected in some way (such as change over time) .You are learning facts about dogs, and each day you do a short test to see how good you are. These are the results:

Table:Facts I got Correct
Day 1	Day 2	Day 3	Day 4
3	4	12	15

And here is the same data as a Line Graph:

You seem to be improving!

You can create graphs like that using our Data Graphs (Bar, Line and Pie) page.

Or we can draw them ourself!

Let's draw a Line Graph for this data:

Table:Ice Cream Sales
Mon	Tue	Wed	Thu	Fri	Sat	Sun
$410	$440	$550	$420	$610	$790	$770

And let's make the vertical scale go from $0 to $800, with tick marks every $200

Draw a vertical scale with tick marks

Label the tick marks, and give the scale a label

Draw a horizontal scale with tick marks and labels

Put a dot for each data value

Connect the dots and give the graph a title

Make sure to have:

·         Vertical scale with tick marks and labels

·         Horizontal scale with tick marks and labels

·         Data points connected by lines

·         A Title

F.      Pictographs

A Pictograph is a way of showing data using images. Each image stands for a certain number of things.

Example: Apples Sold

Here is a pictograph of how many apples were sold at the local shop over 4 months:

Note that each picture of an apple means 10 apples (and the half-apple picture means 5 apples).

So the pictograph is showing:

·         In January 10 apples were sold

·         In February 40 apples were sold

·         In March 25 apples were sold

·         In April 20 apples were sold

It is a fun and interesting way to show data.

But it is not very accurate: in the example above we can't show just 1 apple sold, or 2 apples sold etc.

Why don't you try to make your own pictographs? Here are a few ideas:

·         How much money you have (week by week)

·         How much exercise you get (each day)

·         How many hours you watch TV every week

·         How many sports stories are in each newspaper

G.    Pie Chart

Pie Chart: a special chart that uses "pie slices" to show relative sizes of data.

Imagine you survey your friends to find the kind of movie they like best:

Table:Favorite Type of Movie
Comedy	Action	Romance	Drama	SciFi
4	5	6	1	4

You can show the data by this Pie Chart:

It is a really good way to show relative sizes: it is easy to see which movie types are most liked, and which are least liked, at a glance.

You can create graphs like that using our Data Graphs (Bar, Line and Pie) page.

Or you can make them yourself ...

How to Make Them Yourself

First, put your data into a table (like above), then add up all the values to get a total:

Table:Favorite Type of Movie
Comedy	Action	Romance	Drama	SciFi	TOTAL
4	5	6	1	4	20

Next, divide each value by the total and multiply by 100 to get a percent:

Comedy	Action	Romance	Drama	SciFi	TOTAL
4	5	6	1	4	20
4/20 = 20%	5/20 = 25%	6/20 = 30%	1/20 = 5%	4/20 = 20%	100%

Now to figure out how many degrees for each "pie slice" (correctly called a sector).

A Full Circle has 360 degrees, so we do this calculation:

Comedy	Action	Romance	Drama	SciFi	TOTAL
4	5	6	1	4	20
20%	25%	30%	5%	20%	100%
4/20 × 360° = 72°	5/20 × 360° = 90°	6/20 × 360° = 108°	1/20 × 360° = 18°	4/20 × 360° = 72°	360°

 

Now you are ready to start drawing!

Draw a circle.

Then use your protractor to measure the degrees of each sector.

Here I show the first sector ...

Finish up by coloring each sector and giving it a label like "Comedy: 4 (20%)", etc.

(And dont forget a title!)

Another Example

You can use pie charts to show the relative sizes of many things, such as:

·         what type of car people have,

·         how many customers a shop has on different days and so on.

·         how popular are different breeds of dogs

Example: Student Grades

Here is how many students got each grade in the recent test:

A	B	C	D
4	12	10	2

And here is the pie chart:

Circle Sector and Segment

Slices

There are two main "slices" of a circle:

·         The "pizza" slice is called a Sector.

·         And the Segment, which is cut from the circle by a "chord" (a line between two points on the circle).

Try Them!

Sector	Segment

Common Sectors

The Quadrant and Semicircle are two special types of Sector:

Half a circle is  aSemicircle.

Quarter of a circle is
a Quadrant.

Area of a Sector

You can work out the Area of a Sector by comparing its angle to the angle of a full circle.

Note: we are using radians for the angles.

This is the reasoning:

A circle has an angle of 2π and an Area of:	πr2
A Sector with an angle of θ (instead of 2π) has an Area of:	(θ/2π) × πr2
Which can be simplified to:	(θ/2) × r2

 Area of Sector = θ 2 × r²   (when θ is in radians)

Area of Sector = θ × π360 × r² (when θ is in degrees)

Area of Segment

The Area of a Segment is the area of a sector minus the triangular piece (shown in light blue here).

There is a lengthy reason, but the result is a slight modification of the Sector formula:

Area of Segment = θ − sin(θ)2 × r²   (when θ is in radians)

Area of Segment = ( θ360 × π − sin(θ)2 ) × r²   (when θ is in degrees)

Arc Length The arc length (of a Sector or Segment) is: L = θ × r   (when θ is in radians) L = (θ × π/180) × r   (when θ is in degrees)

H.    Scatter Plots

A Scatter (XY) Plot has points that show the relationship between two sets of data. In this example, each dot shows one person's weight versus their height. (The data is plotted on the graph as "Cartesian (x,y) Coordinates")

Example:

The local ice cream shop keeps track of how much ice cream they sell versus the noon temperature on that day. Here are their figures for the last 12 days:

Ice Cream Sales vs Temperature
Temperature °C	Ice Cream Sales
14,2°	$215
16,4°	$325
11,9°	$185
15,2°	$332
18,5°	$406
22,1°	$522
19,4°	$412
25,1°	$614
23,4°	$544
18,1°	$421
22,6°	$445
17,2°	$408

And here is the same data as a Scatter Plot:

It is now easy to see that warmer weather leads to more sales, but the relationship is not perfect.

Line of Best Fit

We can also draw a "Line of Best Fit" (also called a "Trend Line") on our scatter plot:

Try to have the line as close as possible to all points, and as many points above the line as below.

Example: Sea Level Rise

A Scatter Plot of Sea Level Rise:
And here I have drawn on a "Line of Best Fit".

Interpolation and Extrapolation

Interpolation is where we find a value inside our set of data points.

Here we use linear interpolation to estimate the sales at 21 °C.

Extrapolation is where we find a value outside our set of data points.

Here we use linear extrapolation to estimate the sales at 29 °C (which is higher than any value we have).

Careful: Extrapolation can give misleading results because we are in "uncharted territory".

As well as using a graph (like above) we can create a formula to help us.

Example:

We can estimate a straight line equation from two points from the graph above

Let's estimate two points on the line near actual values: (12°, $180) and (25°, $610)

First, find the slope:

slope "m"	= change in ychange in x
	= $610 − $18025° − 12°
	=  $43013°
	= 33 (rounded)

Now put the slope and the point (12°, $180) into the "point-slope" formula:

y − y₁ = m(x − x₁)

y − 180 = 33(x − 12)

y = 33(x − 12) + 180

y = 33x − 396 + 180

y = 33x − 216

Now we can use that equation to interpolate a sales value at 21°:

y = 33×21 − 216 = $477

And to extrapolate a sales value at 29°:

y = 33×29 − 216 = $741

The values are close to what we got on the graph. But that doesn't mean they are more (or less) accurate. They are all just estimates.

Don't use extrapolation too far! What sales would you expect at 0° ?

y = 33×0 − 216 = −$216

Hmmm... Minus $216? We extrapolated too far!

Note: we used linear (based on a line) interpolation and extrapolation, but there are many other types, for example we could use polynomials to make curvy lines, etc.

Correlation

When the two sets of data are strongly linked together we say they have a High Correlation.

The word Correlation is made of Co- (meaning "together"), and Relation

·         Correlation is Positive when the values increase together, and

·         Correlation is Negative when one value decreases as the other increases

Like this:

(Learn More About Correlation)

Negative Correlation

Correlations can be negative, which means there is a correlation but one value goes down as the other value increases.

Example : Birth Rate vs Income The birth rate tends to be lower in richer countries. Below is a scatter plot for about 100 different countries.

Country Yearly Production per Person Birth Rate Madagascar $800 5,70 India $3.100 2,85 Mexico $9.600 2,49 Taiwan $25.300 1,57 Norway $40.000 1,78

It has a negative correlation (the line slopes down)

Note: I tried to fit a straight line to the data, but maybe a curve would work better, what do you think?

I.       Stem and Leaf Plots

A Stem and Leaf Plot is a special table where each data value is split into a "stem" (the first digit or digits) and a "leaf" (usually the last digit). Like in this example:

Example:

"32" is split into "3" (stem) and "2" (leaf).

Stem "1" Leaf "5" means 15

The "stem" values are listed down, and the "leaf" values go right (or left) from the stem values.

The "stem" is used to group the scores and each "leaf" shows the individual scores within each group.

Example: Long Jump

Sam got his friends to do a long jump and got these results:

2,3, 2,5, 2,5, 2,7, 2,8 3,2, 3,6, 3,6, 4,5, 5,0

And here is the stem-and-leaf plot:

Stem	Leaf
2	3 5 5 7 8
3	2 6 6
4	5
5	0

Stem "2" Leaf "3" means 2,3

Note:

·         Say what the stem and leaf mean (Stem "2" Leaf "3" means 2,3)

·         In this case each leaf is a decimal

·         It is OK to repeat a leaf value

·         5,0 has a leaf of "0"

2.3  Central Value, Mean, Median, Mode, and Outliers

A.    Finding a Central Value

When you have two or more numbers it is nice to find a value for the "center".

a.       2  Numbers

With just 2 numbers the answer is easy: go half-way between.

Example: what is the central value for 3 and 7?

Answer: Half-way between, which is 5.

You can calculate it by adding 3 and 7 and then dividing the result by 2:

(3+7) / 2 = 10/2 = 5

b.      3 or More Numbers

You can use the same idea when you have 3 or more numbers:

Example: what is the central value of 3, 7 and 8?

Answer: You calculate it by adding 3, 7 and 8 and then dividing the results by 3 (because there are 3 numbers):

(3+7+8) / 3 = 18/3 = 6

Notice that we divided by 3 because we had 3 numbers ... very important!

B.     The Mean

So far we have been calculating the Mean (or the Average):

Mean: Add up the numbers and divide by how many numbers.

But sometimes the Mean can let you down:

Example1 : Birthday Activities

Uncle Bob wants to know the average age at the party, to choose an activity.

There will be 6 kids aged 13, and also 5 babies aged 1.

Add up all the ages, and divide by 11 (because there are 11 numbers):

(13+13+13+13+13+13+1+1+1+1+1) / 11 = 7,5...

The mean age is about 7½, so he gets a Jumping Castle! The 13 year olds are embarrassed, and the 1 year olds can't jump!

The Mean was accurate, but in this case it was not useful.

Example 2 : What is the Mean of these numbers?

6, 11, 7

·         Add the numbers: 6 + 11 + 7 = 24

·         Divide by how many numbers (there are 3 numbers): 24 / 3 = 8

The Mean is 8

It is because 6, 11 and 7 added together is the same as 3 lots of 8:

It is like you are "flattening out" the numbers

Example 3 : Look at these numbers:

3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29

The sum of these numbers is 330

There are fifteen numbers.

The mean is equal to 330 / 15 = 22

The mean of the above numbers is 22

How do you handle negative numbers? Adding a negative number is the same as subtracting the number (without the negative). For example 3 + (−2) = 3−2 = 1.

Knowing this, let us try an example:

Example 4 : Find the mean of these numbers:

3, −7, 5, 13, −2

• The sum of these numbers is 3 − 7 + 5 + 13 − 2 = 12
• There are 5 numbers.
• The mean is equal to 12 ÷ 5 = 2,4

The mean of the above numbers is 2,4

Here is how to do it one line:

Mean =	3 − 7 + 5 + 13 – 2	=	12	=  2,4
	5		5

C.    The Median

But you could also use the Median: simply list all numbers in order and choose the middle one:

Example: Birthday Activities (continued)

List the ages in order:

1, 1, 1, 1, 1, 13, 13, 13, 13, 13, 13

Choose the middle number:

1, 1, 1, 1, 1, 13, 13, 13, 13, 13, 13

The Median age is 13 ... so let's have a Disco!

Sometimes there are two middle numbers. Just average them:

Example: What is the Median of 3, 4, 7, 9, 12, 15

There are two numbers in the middle:

3, 4, 7, 9, 12, 15

So we average them:

(7+9) / 2 = 16/2 = 8

The Median is 8                                  

Example: find the Median of 12, 3 and 5

Put them in order:

3, 5, 12

The middle is 5, so the median is 5.

Example:

3, 13, 7, 5, 21, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29

When we put those numbers in order we have:

3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 39, 40, 56

There are fifteen numbers. Our middle is the eighth number:

3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 39, 40, 56

 The median value of this set of numbers is 23.

(It doesn't matter that some numbers are the same in the list.)

BUT, with an even amount of numbersthings are slightly different.In that case we find the middle pair of numbers, and then find the value that is half way between them. This is easily done by adding them together and dividing by two.

Example:

3, 13, 7, 5, 21, 23, 23, 40, 23, 14, 12, 56, 23, 29

When we put those numbers in order we have:

3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 40, 56

There are now fourteen numbers and so we don't have just one middle number, we have a pair of middle numbers:

3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 40, 56

In this example the middle numbers are 21 and 23.

To find the value halfway between them, add them together and divide by 2:

21 + 23 = 44
then 44 ÷ 2 = 22

So the Median in this example is 22.

D.    The Mode

Mode is the value that occurs most often:

Example: Birthday Activities (continued)

Group the numbers so we can count them:

1, 1, 1, 1, 1, 13, 13, 13, 13, 13, 13

"13" occurs 6 times, "1" occurs only 5 times, so the mode is 13.

How to remember? Think "mode is most"

But Mode can be tricky, there can sometimes be more than one Mode.

Example: What is the Mode of 3, 4, 4, 5, 6, 6, 7

Well ... 4 occurs twice but 6 also occurs twice.

So both 4 and 6 are modes.

When there are two modes it is called "bimodal", when there are three or more modes we call it "multimodal".

Example:

3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29

In order these numbers are:

3, 5, 7, 12, 13, 14, 20, 23, 23, 23, 23, 29, 39, 40, 56

This makes it easy to see which numbers appear most often.

In this case the mode is 23.

Another Example: {19, 8, 29, 35, 19, 28, 15}

Arrange them in order: {8, 15, 19, 19, 28, 29, 35}

19 appears twice, all the rest appear only once, so 19 is the mode.

·         More Than One Mode

We can have more than one mode.

Example: {1, 3, 3, 3, 4, 4, 6, 6, 6, 9}

3 appears three times, as does 6.

So there are two modes: at 3 and 6

Having two modes is called "bimodal".

Having more than two modes is called "multimodal".

·         Grouping

When all values appear the same number of times the idea of a mode is not useful. But we could group them to see if one group has more than the others.

Example: {4, 7, 11, 16, 20, 22, 25, 26, 33}

Each value occurs once, so let us try to group them.

We can try groups of 10:

• 0-9: 2 values (4 and 7)
• 10-19: 2 values (11 and 16)
• 20-29: 4 values (20, 22, 25 and 26)
• 30-39: 1 value (33)

In groups of 10, the "20s" appear most often, so we could choose 25 as the mode.

E.     Outliers

Outliersare values that "lieoutside" the other values.

They can change the mean a lot, so we can either not use them (and say so) or use the median or mode instead.

Example: 3, 4, 4, 5 and 104

Mean: Add them up, and divide by 5 (as there are 5 numbers):

(3+4+4+5+104) / 5 = 24

24 doesn't seem to represent those numbers well at all!

Without the 104 the mean is:

            (3+4+4+5) / 4 = 4

But do tell people you are not including the 104.

Median: They are in order, so just choose the middle number, which is 4:

3, 4, 4, 5, 104

Mode: 4 occurs most often, so the Mode is 4

3, 4, 4, 5, 104

REFERENCES

www.mathisfun.com