Data Description.pdf

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Data Description

MTK3006

Department of Mathematics

Faculty of Science and Technology

Universiti Malaysia Terengganu

[email protected]

MTK3006 Statistics for Chemists Data Description
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Part I

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Basic Terms

Population

A population is a collection of all subjects or objects of interest.



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Basic Terms

Population

A population is a collection of all subjects or objects of interest.

Sample

A sample is a portion or part of the population of interest.



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Basic Terms

Variable

A variable is a characteristic or attribute that can assume different values.

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Basic Terms

Variable


Data

The values that a variable can assume are called data.



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Basic Terms

Variable


Data


Data set

A collection of data values or measurements forms a data set.



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Basic Terms

Variable


Data


Data set

A collection of data values or measurements forms a data set.

Types of data

Quantitative data is a numerical measurement expressed in terms ofnumbers.

Qualitative data is a categorical measurement expressed by means of anatural language description.



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Basic Terms

Parameter

A parameter is a characteristic or measure obtained by using all the datavalues from a population.

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Basic Terms

Parameter

A parameter is a characteristic or measure obtained by using all the datavalues from a population.

Statistic

A statistic is a characteristic or measure obtained by using the data valuesfrom a sample.

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Basic Terms

Statistics

Statistics is the science of collecting, organizing, summarizing, analyzingand interpreting data.

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Basic Terms

Statistics

Statistics is the science of collecting, organizing, summarizing, analyzingand interpreting data.

Areas of statistics

The branch of statistics devoted to the organization, summarization,description and presentation of data sets is called descriptive statistics.

The branch of statistics concerned with using sample data to drawconclusions about a population is called inferential statistics.

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Part II

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Describing Data with Tables

Data collected in original form is called raw data.

A frequency distribution is the organization of raw data in table form

using classes and frequencies. There are three types of frequency distributions:

Categorical frequency distributions Ungrouped frequency distributions Grouped frequency distributions



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Categorical Frequency Distribution

Can be used for data that can beplaced in specific categories.

Examples political affiliation,

religious affiliation, blood type, etc. Example Blood Type Data

A,B,B,AB,O,O,O,B,AB,B,B,B,O,A,O,A,O,O,O,AB,AB,A,O,B,A

Blood Type Frequency Distribution

Class Frequency Percent

A 5 20B 7 28O 9 36

AB 4 16

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Ungrouped Frequency Distribution

Can be used for data that can beenumerated and when the range ofvalues in the data set is not large.

Examples number of kilometersyour instructors have to travel fromhome to campus, number of girls in4-child family, etc.

Example Number of Kilometers

Travelled: 8, 5, 6, 5, 5, 7, 7

Number of Kilometers Travelled

Class Frequency

5 36 17 28 1


G d F Di ib i
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Grouped Frequency Distribution

Can be used when the range of values in the data set is very large.

Class limits represent the smallest and largest data values that can beincluded in a class. The smallest and largest possible data values in aclass are the lower and upper class limits.

Class boundaries separate the classes. To find a class boundary,average the upper class limit of one class and the lower class limit ofthe next class.

The class width is found by subtracting the lower (or upper) class limitof one class from the lower (or upper) class limit of the previous class.

The class midpoint can be calculated by averaging the upper and lowerclass limits.


G d F Di ib i
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Rules for classes

There should be 5-20 classes.

The class width should be an odd number.

The classes must not overlap.

The classes must not have breaks.

The classes must include all the data values.

The classes must be equal in width.


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To construct a grouped frequency distribution:

Find the highest and lowest values.

Find the range.

Choose the number of classes.

Find the class width by dividing the range by the number ofclasses and rounding up.

Choose a starting point (usually the lowest value); add the classwidth to get all the lower limits.

Find the upper class limits.

Find the class boundaries.

Find the frequencies and the cumulative frequencies.


G d F Di t ib ti
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Construct a grouped frequency distribution using 7 classes.

112 100 127 120 134 118 105 110 109 112 110118 117 116 118 122 114 114 105 109 107 112114 115 118 117 118 122 106 110 116 108 110

121 113 120 119 111 104 111 120 113 120 117105 110 118 112 114 114


G d F Di t ib ti
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Construct a grouped frequency distribution using 7 classes.

112 100 127 120 134 118 105 110 109 112 110118 117 116 118 122 114 114 105 109 107 112114 115 118 117 118 122 106 110 116 108 110

121 113 120 119 111 104 111 120 113 120 117105 110 118 112 114 114

Class Limits Class Boundaries Frequency Cumulative Frequency100 - 104 99.5 - 104.5 2 2


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Class Limits Class Boundaries Frequency Cumulative Frequency100 - 104 99.5 - 104.5 2 2105 - 109 104.5 - 109.5 8 10

110 - 114 109.5 - 114.5 18 28115 - 119 114.5 - 119.5 13 41120 - 124 119.5 - 124.5 7 48125 - 129 124.5 - 129.5 1 49130 - 134 129.5 - 134.5 1 50

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Part III


Measures of Central Tendency
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Measures of Central Tendency

Given a set of data, we often would like to have one number that isrepresentative of a population or sample.

There are several standard ways to measure the center. Meanthe average of the data set

Medianthe midpoint of the data set

Modethe value that occurs most often in the data set


The Mean


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The Mean

Denote by xi the ith observed data value in the population or sample.

Denote by N and n the population and sample sizes respectively. The population mean is the sum of all the population values divided by the

total number of population values:

=

1

N

Ni=1

xi.

The sample mean is the sum of all the sample values divided by the number ofsample values:

x = 1n

ni=1

xi.

Find the sample mean of 20, 26, 40, 36, 23, 42, 35, 24, 30.


The Mean


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The Mean

Denote by xi the ith observed data value in the population or sample.

Denote by N and n the population and sample sizes respectively. The population mean is the sum of all the population values divided by the

total number of population values:

=

1

N

Ni=1

xi.

The sample mean is the sum of all the sample values divided by the number ofsample values:

x = 1n

ni=1

xi.

Find the sample mean of 20, 26, 40, 36, 23, 42, 35, 24, 30.Answer: x = 30.67


The Median
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The Median

The median is the middle value, or the average of the middle twovalues, of a population or sample, when the data values are arrangedfrom smallest to largest.

The median will be one of the data values if there is an odd numberof values.

The median will be the average of two data values if there is an evennumber of values.

Find the median of 684, 764, 656, 702, 856, 1133, 1132, 1303.


The Median
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The Median

The median is the middle value, or the average of the middle twovalues, of a population or sample, when the data values are arrangedfrom smallest to largest.

The median will be one of the data values if there is an odd numberof values.

The median will be the average of two data values if there is an evennumber of values.

Find the median of 684, 764, 656, 702, 856, 1133, 1132, 1303.

Answer: Median = 810


The Mode
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The Mode

The mode is the value in the population or sample that occurs mostfrequently.

It is sometimes said to be the most typical case.

There may be no mode, one mode (unimodal), two modes (bimodal),or many modes (multimodal).

Find the mode of 18.0, 14.0, 34.5, 10, 11.3, 10, 12.4, 10.

Find the mode of 104, 104, 104, 104, 104, 107, 109, 109, 109, 110,

109, 111, 112, 111, 109.


Properties of the Mean
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Properties of the Mean

Uses all data values.

Sample mean varies less than the sample median or mode.

Used in computing other statistics, such as the variance. Unique, usually not one of the data values.

Affected by extremely high or low values, called outliers.


Properties of the Median
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Properties of the Median

Gives the midpoint.

Used when it is necessary to find out whether the data values fall into

the upper half or lower half of the data set. Affected less than the mean by extremely high or extremely low

values.


Properties of the Mode
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p

Used when the most typical case is desired.

Easiest to compute. Not always unique or may not exist.


Measures of Dispersion
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p

Dispersion refers to the spread or variability in a data set.

Measures of dispersion include range, variance, standard deviation,etc.


The Range
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g

The range is the difference between the highest and lowest values of a

population or sample. Two experimental brands of outdoor paint are tested to see how long

each will last before fading. Six cans of each brand constitute a smallpopulation. The results (in months) are:

Brand A Brand B

10 35

60 45

50 30

30 35

40 4020 25

The population mean for both brands is the same.

Which brand would you buy?


The Variance
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The variance is the average of the squares of the distance each value

is from the mean. The population variance is

2 =

1

N

N

i=1

(xi )2.

The sample variance is

s2 =1

n

1

n

i=1

(xi x)2.

This formula for s2 makes a better estimator of2 than if we haddivided by n.


The Standard Deviation
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The standard deviation is the square root of the variance.

The population standard deviation is .

The sample standard deviation is s.

The standard deviation is measured in the same unit as themeasurements in the population or sample.

A large standard deviation indicates that the data values are far fromthe mean, whereas a small standard deviation indicates that they are

clustered closely around the mean.


Alternate Formula for the Sample Standard Deviation
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s =

1

n 1

n

i=1x2

i

1

n

n

i=1xi

2

Saves time when calculating by hand.

Does not use the sample mean.

Find the sample standard deviation of 11.2, 11.9, 12.0, 12.8, 13.4,14.3.


Alternate Formula for the Sample Standard Deviation
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s =

1

n 1

n

i=1x2

i

1

n

n

i=1xi

2

Saves time when calculating by hand.

Does not use the sample mean.

Find the sample standard deviation of 11.2, 11.9, 12.0, 12.8, 13.4,14.3. Answer: s = 1.13


Measures of Position
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Measures of position or location are used to locate the relativeposition of a data value in the data set.

These measures include: z-score quartiles outlier


The z-score
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A z-score or standard score for a value is obtained by subtracting themean from the value and dividing the result by the standard deviation.

The formula for the population (or sample) z-score is

z =x

or =

x x

s

.

A z-score represents the number of standard deviations a value isabove or below the mean.


The Quartiles
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Quartiles separate the data set into 4 equal groups.

The first quartile (Q1) is the value that lies 25% of the way up fromthe smallest value.

The second quartile (Q2) is the value that lies 50% of the way upfrom the smallest value, and is equivalent to the median.

The third quartile (Q3) is the value that lies 75% of the way up fromthe smallest value.

The interquartile range (IQR) is the difference between the upper and

lower quartiles, i.e., IQR = Q3 Q1.


The Outlier
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An outlier is an extremely high or low data value when compared withthe rest of the data values.

A data value less than Q1 1.5 IQR or greater thanQ3 + 1.5 IQR can be considered an outlier.

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Part IV


Describing Data with Graphs
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Graphs used for qualitative data

Bar charts Pareto charts

Graphs used for quantitative data

Histograms Frequency polygons Stem and leaf plots Box plots Time series plots


Bar Chart
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A bar chart is a chart withrectangular bars.

The bars can be plottedvertically or horizontally.

Example Modes ofTransportation to Work

The vertical scale showsfrequencies.

The horizontal scale shows

categories.

How people get to work

People

0

5

10

15

20

25

30

Car Bus Train Walk


Pareto Chart
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A Pareto chart can be used to

represent a categorical frequencydistribution. It is a bar chartarranged in descending order ofheight from left to right.

How people get to work

People

0

5

10

15

20

25

30

Car Train Bus Walk


Histogram
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The histogram is a graph that displays the quantitative data by usingvertical bars of various heights to represent the frequencies of the

classes. The histogram is similar to the bar chart, but it is drawn without gaps

between the bars. The class boundaries are represented on the horizontal axis.

Record High Temperatures

Temperature ( F)

Frequency

99.5 104.5 109.5 114.5 119.5 124.5 129.5 134.5

0

3

6

9

12

15

18

| | | | | | | |


Frequency Polygon
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The frequency polygon is a graph that displays the quantitative databy using lines that connect points plotted for the frequencies at the

class midpoints. The frequencies are represented by the heights of the points. The class midpoints are represented on the horizontal axis.

q

q

q

q

q

q

q qq

Record High Temperatures

Temperature ( F)

Frequency

102 107 112 117 122 127 1320

3

6

9

12

15

18

| | | | | | |


Stem and Leaf Plot
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A stem and leaf plot is a data plot that uses part of a data value asthe stem and part of the data value as the leaf to form groups orclasses.

In a stem and leaf plot, each data value is split into a stem and a leaf.

The leaf is usually the last digit of the data value and the other digits

to the left of the leaf form the stem.

For example, the number 123 would be split as:

stem 12leaf 3

The stems are listed on the left and the corresponding leaves on theright.


Stem and Leaf Plot
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Construct a stem and leaf plot.

25 31 20 32 1314 43 2 57 2336 32 33 32 4432 52 44 51 45


Stem and Leaf Plot
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Construct a stem and leaf plot.

25 31 20 32 1314 43 2 57 2336 32 33 32 4432 52 44 51 45

0 21 3 4

2 0 3 53 1 2 2 2 2 3 64 3 4 4 55 1 2 7


Box Plot
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A box plot is a graph that presents information from a five-numbersummary.

The five-number summary is composed of the minimum, Q1, median,Q3 and maximum.

The five-number summary can be graphically represented by using abox plot.


Box Plot
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To construct a box plot:

Find the five-number summary.

Draw a horizontal axis with a scale that includes the maximum andminimum data values.

Draw a box with vertical sides through Q1 and Q3, and draw avertical line though the median.

Draw a line from the minimum data value to the left side of the boxand a line from the maximum data value to the right side of the box.


Box Plot
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Construct a box plot for the data:89, 47, 164, 296, 30, 215, 138, 78, 48, 39


Box Plot
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Five-number summary30-47-83.5-164-296


Box Plot
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Five-number summary30-47-83.5-164-296

0 100 200 300

30

47 83.5 164

296


Time Series Plot
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A time series plot represents data that occur over a specific period oftime.

It is a line graph where the time is represented on the horizontal axisand the quantity that varies over time is represented on the verticalaxis.

q

qq

q

q

q

q

q

qq

Temperature over a 9Hour Period

Time

Temperature(

F)

1 2 3 4 5 6 7 8 912

35

40

45

50

55

60

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Part V


R and R Commander
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R

A language and environment for statistical computing and graphics

Available as a free software at http://www.r-project.org/

A command-driven statistical program

R Commander

A graphical user interface for R

Its interface includes menus, buttons and a few other elements

http://www.r-project.org/http://www.r-project.org/http://find/

Documents

Data Description.pdf