Statistics for math (English Version)

STATISTICSSTATISTICS

By: Tito Nurdiyantotnk_adhyastha@yahoo.com

Qualitative Data Quantitative Data

Overview:Deals with descriptions.Data can be observed but not measured.Colors, textures, smells, tastes, appearance, beauty, etc.Qualitative → QualityExamples :1.Robust arom2.Frothy appearance3.Strong taste4.Cup

Overview: Deals with numbers.Data which can be measured.Length, height, area, volume, weight, speed, time, temperature, humidity, sound levels, cost, members, ages, etc.Quantitative → Quantity Examples :1.10 ounces of latte2.Serving temperature 140 F3.Serving cup 6 inches in height4.Cost $ 6

Differences between Differences between Qualitative and Quantative Qualitative and Quantative DataData

Please write in a piece of Please write in a piece of paper yours:paper yours:

1. AGE

2. SUBJECT THAT YOU LIKE

3. HEIGHT

4. WEIGHT

LETS SEE THE RESULTLETS SEE THE RESULT

AGE

13 YEARS OLD : ………0……… PERSON

14 YEARS OLD : ………0……… PERSON

15 YEARS OLD : ………6…… PERSON

16 YEARS OLD : ………17…… PERSON

17 YEARS OLD : ………1…… PERSON

Lets we make a display of Lets we make a display of your data!!!your data!!!What kind of display that What kind of display that we will we will made??????????????????made??????????????????

We can display using BAR We can display using BAR CHART and PIE CHARTCHART and PIE CHART

Reading and Reading and Representing DataRepresenting DataThere are many ways to presenting data, here they are

PictogramPictogram

Pie chartPie chart

Mohammed asked his class about their favorite flavour of ice cream.

Vanilla 12

Strawberry 6

Chocolate 5

Other 7

Their answers were:

He decided to draw a pie chart of his He decided to draw a pie chart of his resultsresults

Flavour Number of pupils

Working Angle Percentage

Vanilla 12

Strawberry 6

Chocolate 5

Other 7

--

12360

30

6360

30

5360

30

7360

30

30

144

72

60

84

360

Use the table to help you draw a pie chart. Don’t forget the key

Vanilla

Strawberry

Chocolate

Other

144

72

60

84

Compound Bar Chart

Vertical Bar Chart Horizontal Bar Chart

Bar ChartBar Chart

Line DiagramLine DiagramIt is usual to present data that you can find in newspaper, magazine and the others media. You right, it is useful in the fields of statistics and science, are one of the most common tools used to present data. So, why line graph popular? Line graph are more popular than all graph because their visual characteristics reveal data trends clearly and these graph are easy to create. Although they do not present specific data as well as tables do but line graphs are able to show relationships more clearly than tables do. Here is the example line graph.

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10 12

Number of chocolate bars

Nu

mb

er

of

pu

pils

School A

School B

How many pupils from school A ate 7 bars a week?

How many pupils from school B ate 7 bars a week?

How many pupils from A ate no chocolate?

Which set of pupils ate the most chocolate?

HistogHistogramramSalary of number of employees a factory, shown the diagram below.

•In a histogram, frequency is measured by the area of the column. •In a vertical bar graph, frequency is measured by the height of the bar.

Polygon is a graph formed by joining the midpoints of histogram column tops. These graphs are used only when depicting data from the continuous variables shown on a histogram. It is useful demonstrating continuity of the variable being studied. Here is good example polygon. Here is the polygon of salary the number of employees in a factory.

PolygonPolygon

Dot DiagramDot DiagramThe dot diagram displays information using dot. Each dot represent a piece of data along a scale with frequency represented on the other scale.

QUARTILE (singular data)

• The median of a set of data separates the data into two equal parts.   •The first quartile is the median of the lower part of the data.•The second quartile is another name for the median of the entire set of data.• The third quartile is the median of the upper part of the data.

If n is odd, the median is the value.

If n is even, the median is halfway the value and the following value

1222

1nn xx

2

1nx

The Measure of Central Tendency

EXAMPLE (QUARTILE)

Math test score 80, 75, 90, 95, 65, 65, 80, 85, 70, 100. I will represent the data in a box and whisker plot. Could you help me?

Write the data in numerical order and find the first quartile, the median, the third quartile, the smallest value and the largest value.•median = 80•first quartile = 70•third quartile = 90•smallest value = 65largest value = 100

MODEThe mode is the most frequently occurring score. In other word, the mode is simply the number which appears most often. Sometimes mode is also called modal value

Example The number of points scored in a series of football games is listed below. Which score occurred often? 6, 3, 9, 5, 13, 15, 9, 18, 9, 8Answer: Ordering the scores from least to greatest, we get: 3, 5, 6, 8, 9, 9, 9, 13, 15, 18Hi.., now we can see easier that the score which occurs most often is 9

The mode of a set of data is the value in the set that occurs most

often. A set of data can be bimodal. It is also possible to have

a set of data with no mode

Remember!!

On a cold winter day in January, the temperature for 9 North American cities is recorded in Fahrenheit. What is the mode of these temperatures? -9, 0, -2, 3, 5, -1, 10, 4, 6

Answer:Ordering data from least to greatest, we get-9, -2, -1, 0, 3, 4, 5, 6, 10Since each value occurs only once in the data set, there is no mode for this set of data.

MEDIAN

Example

If n is odd, the median is the value.

If n is even, the median is halfway the value and the following value

2

1nx

122

nn xx

A marathon race was completed by 5 participants. What is the median of these times given in hours?

1.5 hr, 4.2 hr, 3.5 hr, 6.7 hr 5.0 hr

The number of data is odd and the total number of data is 5 so that:Median

2.45

12

1

XXn

MEAN

Example

Mean is the average of a series score. It is meant that the sum of the values in the data set divided by the number of values. So, we can represent mean with this formula ;

n

fx

n

xxxxx

n

ii

n

1

321 ...

1. Find mean for the following data : 5, 3, 4, 6, 7 Answer :

55

25

5

76435

54321

x

n

xxxxxx

Example

2. A class contain 20 men and 30 women. If men’s average heightis 167 and women’s average height is 158 cm, then find the average height students in that class.

Answer: The average height =

6,1613020

)158()30()167()20(

cmcmx

x

DESIL

Datum yang membagi data terurut menjadi sepersuluh bagian

Menentukan Dm

•Hitung m(n+1) /10• Jika hasilnya bulat misal r maka Dm

= xm

• Jika hasilnya Ganjil ( bukan bulat) tapi terletak antara r dan r+1 maka Dm = Xm + (m(n+1)/10 – r ) (Xr+1 – Xr )

x

Contoh : 7, 5, 6, 5, 3, 6, 4, 8, 2, 6, 8, 7 Tentukan D1

Peny: 2, 3, 4, 5, 5, 6, 6,6 , 7,7, 8,8 D1

= 1 (12 + 1) / 10 = 1,3

D1 = X1 + ( 1,3 – 1) (X2 – X1) = 2 + (0,3)(3-2) = 2 + 0,3 = 2,3

Example

3. A family must drive an average 250 miles per day to complete their Vacation on time. On the first five days, they travel 220 miles, 300 miles, 210 miles, 275 miles and 240 miles. How many miles must they travel on the sixth day in order to finish their vacation on time?

x

n

fx

n

ii

1

The sum of the first 5 days is 1,245 miles. Let

represent the number of miles travelled on the sixth day. We get

x

6

245,1250

x 6250245,1 x

500,1245,1 x

255

245,1500,1

x

x

1. Construct a pie chart of data in one school that the total 100 students that are consist of 50 student like rap music, 25 students like alternative music, 13 students like rock and roll, 10 students like country music and 2 students like classical.

Income (x) in dollars 600 700 800 900 1000

1100

1200

Numbers of workers (f)

30 45 75 36 24 18 6

2. The monthly incomes of 234 workers in a factory are given in the table below. Find the mean of monthly incomes of workers!

Exercise

3. The average height of 50 men is 165,2 cm. If Mr. Susilo is included, then the average height becomes 165,28 cm . Find the height of Mr. Susilo.

Exercise

4. In a crash test, 11 cars were tested to determine what impact speed was required to obtain minimal bumper damage. Find the mode of the speeds given in miles per hour below.

15, 24, 18, 22, 16, 26, 22, 20, 18, 25, 27

5. The test score of 8 eleventh grade students are listed below. Find the median. 82, 92, 75, 94, 85, 100, 89, 78

ANSWER KEY

4.

5.

Ordering the data from least to greatest, we get:75, 78, 82, 85, 89, 92, 94, 100Oops! The number of data is even. So that, the value of median is:Median

872

8985

22

1

n

nXX

Since both 18 and 22 occur twice, the modes are 18 and 22 miles per hour. This data set is bimodal

Number of CD’s

Frequency

0 – 4 10

5 – 9 7

10 – 14 2

15 – 19 4

20 – 24 4

25 – 29 1

30 – 34 2

Ririn asked his class how many CD’s they owned.

Find mode of the following data

6..

ANSWER KEY

1.

Income (x) in dollars

Numbers of workers (f)

x

600 30 18000700 45 31500800 75 60000900 36 324001000 24 240001100 18 198001200 6 7200

235 192.900

ii fx 2.

So, the mean of monthly incomes of workers

35.824235

192900

1

1

k

ii

k

iii

f

fxx

GROUPED DATA

Grouped frequency Distribution

Arrange Cummulative Frequency DistributionArrange Cummulative Frequency DistributionMode

Lower Quartile, Median Quartile, Upper QuartileVarians

Standard Deviation

Mean

Grouped frequency Distribution

Before learning about Grouped Frequency Distribution. Lets see the differences Singular Frequency Distribution below

The Age(xi )

Talley Frequencyfi

10-19 |||| | 620-29 ||| 330-39 |||| 440-49 ||| 350-59 |||| 560-69 |||| | 670-79 ||| 3

The Age(xi )

Talley

Frequencyfi

10 |||| | 6

20 ||| 330 |||| 440 ||| 350 |||| 560 |||| | 6

70 ||| 3

Singular Frequency Distribution Grouped Frequency Distribution

Terminology in Grouped frequency Distribution

1. Class : 1st Class 10-19 , 2nd Class 20-29 etc

The Age(xi )

Frequencyfi

10-19 620-29 330-39 440-49 350-59 560-69 670-79 3

2. Number of Class = 73. Batas kelas ... Batas bawah kelas pertama 10 Batas atas kelas pertama 19

4. Tepi Kelas Tepi bawah = batas bawah - 0,5 Tepi atas = batas atas + 0,55. Panjang Kelas = tepi atas – tepi bawah

It is necessary for us to learn how to find the mean value of group data. In order to present a large set of data more clearly, it can be sorted into groups (or classes).

If the data are representated in classes, then the mode can be found by the formula.

cdd

dL

21

11Mode

1d

2d

1L

MODE

Example (grouped

data)

Mathematics marks of mathematics examination are45 56 30 67 78 49 80 35 85 8765 90 54 78 89 43 78 34 84 5580 93 46 38 93 57 69 35 70 76

This is the way to the data in classes, find the upper, lower class boundary and width of each class

Class Talley Frequency

30-39 |||| | 640-49 ||| 350-59 |||| 460-69 ||| 370-79 |||| 580-89 |||| | 690-100 ||| 3

First it is very easy to you to Make a grouped tally chart for this data.

3940

39.5

49 50

49.5

Between two classes there is a gap, the midpoints of the gaps define the class boundaries

The lower boundary 30-39 is 29.5 and the upper class boundary is of the class is 39.5. The difference between the upper class boundary and the lower class boundary (39.5-29.5=10) is called the class width. Well, could you determine the lower boundary and the upper boundary for each class?

Example (grouped data)

Now, lets begin. The example of mode for grouped data

Find mode of each group

1.

ix if

Mass (x)

(Kg)

Mid Point

Frequency

40-44 42 545-49 47 850-54 52 10

55-59 57 7

ix if

532

25.49

5.51Mode =

Mean grouped data

r

ii

r

iii

f

xf

X

1

1__

.

Example (grouped data)

Marks 51-60 61-70 71-80 81-90 91-100

Number of pupils

25 15 30 22 8

mathematics test taken by 100 Senior High School students:Find the mean of the marks for a mathematics test above!

SOlution

Marks Mid point of marks

Number of pupils

51-60 55.5 25 1387.5

61-70 65.5 15 982.5

71-80 75.5 30 225081-90 85.5 22 188191-100 95.5 8 764

Total = 100

=7265

ix if ii fx

)( ix )( if

i

n

ii xf .

1

n

iif

1

65.72100

7265.

01

1

n

i

n

iii

f

xf

x

c

f

fnLQ

1

1

114

1

c

f

fnLQ

2

2

222

1

c

f

fnLQ

3

3

334

3

1Q

2Q

3Q

lower quartilemedian

upper quartile

iLthe lower boundary of quartile class

n the number of data

if

the number of frequency cummulative before the quartile class

if frequency of quartile class

c

3,2,1i

the length of quartile classAnd

Example

Weight (kg) Frequency

40-49 5

50-59 14

60-69 16

70-79 12

80=89 3

Total 50

The following data represent the weights (in kg) of 50 students

Weight (kg) Frequency Frequency Cumulative

40-49 5 550-59 14 1960-69 16 3570-79 12 4780=89 3 50Total 50

5.12504

1

4

1n

c

f

fnLQ

1

1

114

1

•First Quartile/lower quartile

, ccorrespondence with class 50-59

Solution

86.54

1014

5.75.4910.

14

55.125.49

10.14

5504

1

5.49

1

1

1

Q

Q

Q

25504

2

4

2n

c

f

fnLQ

2

2

222

1

25.63

1016

65.5910.

16

19255.59

10.16

19502

1

5.59

2

2

2

Q

Q

Q

•Second Quartile/Median

correspondence with class 60-69

•Third Quartile/Median

5.37504

3

4

3n

c

f

fnLQ

3

3

334

3

58.71

10.12

355.375.69

10.12

35504

3

5.69

3

3

Q

Q

correspondence with class 70-79

Variance = S2

Standard Deviation = S

2

1

2 1

n

ii xx

nx

For Singular Data

n

ii xx

nx

1

21

ExampleThe height are :156, 160, 158, 166, 168, 170Find out the mean, the variance, and the standard deviation

1636

170168166158160156

1

_

n

iixx

6

163170163168163166163158163160)163156 2222222

6

753537 2222222

67,276

4925925949

So, Standard Deviation is 26.567.272

Variance = S2

Standard Deviation=S

2

1

2 1

n

iii xxf

nx

For Grouped Data

n

iii xxf

nx

1

21

Thank you ^^