modelling and statistical analysis

Amity Business School, Noida

Amity Institute of Environmental

Sciences (AIES)

Semester III

Modelling and Statistical Analysis of Environmental Systems

Ms. Shivangi Somvanshi

[email protected]

1



DATA AND DATA SETS

• Data are the facts and figures collected, summarized, analyzed, and interpreted.

• The data collected in a particular study are The data collected in a particular study are referred to as the referred to as the data setdata set..



The The elementselements are the entities on which data are collected. are the entities on which data are collected.

A A variablevariable is a characteristic of interest for the elements is a characteristic of interest for the elements..

The set of measurements collected for a particular element isThe set of measurements collected for a particular element is called an called an observationobservation..

The total number of data values in a complete data set is the The total number of data values in a complete data set is the number of elements multiplied by the number of variables.number of elements multiplied by the number of variables.

ELEMENTS, VARIABLES, AND ELEMENTS, VARIABLES, AND OBSERVATIONSOBSERVATIONS



Position Position BODBOD DODO THTHSample sitesSample sites(Gomti River)(Gomti River)

Bhatpur (S.P.1)Bhatpur (S.P.1) Gaughat (S.P.2)Gaughat (S.P.2) M.Meakins (S.P.3)M.Meakins (S.P.3) Pipraghat (S.P.4)Pipraghat (S.P.4) Gangaganj (S.P.5)Gangaganj (S.P.5)

USUS 4.2 4.2 6.96.9 184184 USUS 3.4 3.4 7.17.1 198198 DSDS 1212 2.42.4 200 200 USUS 16.5 16.5 1.31.3 208208 DSDS 11 4.311 4.3 170170

VariableVariablessElemenElemen

tt NamesNames

Data SetData Set

ObservatioObservationn



SCALES OF MEASUREMENT

The scale indicates the data summarization and statistical The scale indicates the data summarization and statistical analyses that are most appropriate.analyses that are most appropriate.

The scale determines the amount of information containedThe scale determines the amount of information contained in the data.in the data.

Scales of measurement include:Scales of measurement include:NominalNominalOrdinalOrdinal

IntervalInterval

RatioRatio



1. NOMINAL SCALE

A A non-numeric labelnon-numeric label or or numeric codenumeric code may be used. may be used.

Data are Data are labels or nameslabels or names used to identify an attribute used to identify an attribute of the element.of the element.



2. ORDINAL SCALE

A A non-numeric labelnon-numeric label or or numeric codenumeric code may be used. may be used.

The data have the properties of nominal data andThe data have the properties of nominal data and the the order or rank of the data is meaningfulorder or rank of the data is meaningful..



• Ordinal

Example:Example: Observer provide rating of water quality of the above mentioned sample Observer provide rating of water quality of the above mentioned sample sites as good, poor and very poor. sites as good, poor and very poor. Because the data obtained are the labels—good, poor or very poor—the data have the properties of nominal data. In addition, the data can be ranked, or ordered, with respect to the water quality. Data recorded as good indicate the best quality, followed by poor and very poor. Thus, the scale of measurement is ordinal. Note that the ordinal data canalso be recorded using a numeric code. Code 1 for good, 2 for poor and 3 Code 1 for good, 2 for poor and 3 for very poorfor very poor. .



3. INTERVAL SCALE

Interval data are Interval data are always numericalways numeric..

The data have the properties of ordinal data, andThe data have the properties of ordinal data, and the interval between observations is expressed inthe interval between observations is expressed in terms of a fixed unit of measure.terms of a fixed unit of measure.


Interval

Example:Example: BOD of sample point 1 is 4.2 mg/l and sample BOD of sample point 1 is 4.2 mg/l and sample point 2 is 3.4mg/l. Sample point 1 has more BOD point 2 is 3.4mg/l. Sample point 1 has more BOD than sample point 2 by 0.8mg/l. Similarly sample than sample point 2 by 0.8mg/l. Similarly sample Point 3 has more BOD than sample point 2 by 8.6 mg/l Point 3 has more BOD than sample point 2 by 8.6 mg/l



4. RATIO SCALE

The data have all the properties of interval dataThe data have all the properties of interval data and the and the ratio of two values is meaningfulratio of two values is meaningful..

Variables such as distance, height, weight, and timeVariables such as distance, height, weight, and time use the ratio scale.use the ratio scale.

This This scale must contain a zero valuescale must contain a zero value that indicates that indicates that nothing exists for the variable at the zero point.that nothing exists for the variable at the zero point.



• Ratio

Example:Example: The BOD at a specific location of a River was 6 mg/l The BOD at a specific location of a River was 6 mg/l during pre-monsoon season while it was increased to 12 during pre-monsoon season while it was increased to 12 mg/l during post monsoon season. Ratio shows that the mg/l during post monsoon season. Ratio shows that the BOD during post- monsoon season is double the BOD BOD during post- monsoon season is double the BOD During pre-monsoon season. During pre-monsoon season.



Data can be further classified as being qualitativeData can be further classified as being qualitative or quantitative.or quantitative.

The statistical analysis that is appropriate dependsThe statistical analysis that is appropriate depends on whether the data for the variable are qualitativeon whether the data for the variable are qualitative or quantitative.or quantitative.

In general, there are more alternatives for statisticalIn general, there are more alternatives for statistical analysis when the data are quantitative.analysis when the data are quantitative.

QUALITATIVE AND QUANTITATIVE DATAQUALITATIVE AND QUANTITATIVE DATA



QUALITATIVE DATA Labels or namesLabels or names used to identify an attribute of each used to identify an attribute of each elementelement

Often referred to as Often referred to as categorical datacategorical data

Use either the nominal or ordinal scale of measurementUse either the nominal or ordinal scale of measurement

Can be either numeric or nonnumericCan be either numeric or nonnumeric

Appropriate statistical analyses are rather limitedAppropriate statistical analyses are rather limited



QUANTITATIVE DATAQUANTITATIVE DATA Quantitative data indicate Quantitative data indicate how many or how much:how many or how much:

discretediscrete, if measuring how many, if measuring how many

continuouscontinuous, if measuring how much, if measuring how much

Quantitative data are Quantitative data are always numericalways numeric..

Quantitative data are obtained using either the interval or ratio scale of measurement.



SCALES OF MEASUREMENTSCALES OF MEASUREMENT

QualitativeQualitative QuantitativQuantitativee

NumericalNumerical NumericalNumericalNon-Non-numericalnumerical

DataData

NominaNominall

OrdinaOrdinall

NominalNominal OrdinalOrdinal IntervalInterval RatioRatio



CROSS-SECTIONAL DATACROSS-SECTIONAL DATA

Cross-sectional dataCross-sectional data are collected at the same or are collected at the same or approximately the same point in time.approximately the same point in time.

ExampleExample: Data detailing the water quality parameters : Data detailing the water quality parameters of Gomti River of all the 5 sample points in June 2007.of Gomti River of all the 5 sample points in June 2007.



TIME SERIES DATATIME SERIES DATA

Time series dataTime series data are collected over several time are collected over several time periods.periods.

ExampleExample: Data detailing the water quality parameters : Data detailing the water quality parameters of Gomti River of all the 5 sample points of the last of Gomti River of all the 5 sample points of the last 36 months.36 months.



DESCRIPTIVE STATISTICS

• Descriptive statistics are the tabular, graphical, and numerical methods used to summarize and present data.



TDS (ppm) for 50 samples of a groundwater TDS (ppm) for 50 samples of a groundwater of a districtof a district

91 78 93 57 75 52 99 80 97 6271 69 72 89 66 75 79 75 72 76104 74 62 68 97 105 77 65 80 10985 97 88 68 83 68 71 69 67 7462 82 98 101 79 105 79 69 62 73

91 78 93 57 75 52 99 80 97 6271 69 72 89 66 75 79 75 72 76104 74 62 68 97 105 77 65 80 10985 97 88 68 83 68 71 69 67 7462 82 98 101 79 105 79 69 62 73



NUMERICAL DESCRIPTIVE STATISTICS

TDS of the groundwater of an area , based on the 50TDS of the groundwater of an area , based on the 50 samples studied, is 79ppm (found by summing thesamples studied, is 79ppm (found by summing the 50 TDS values and then dividing by 50).50 TDS values and then dividing by 50).

The most common numerical descriptive statisticThe most common numerical descriptive statistic is the is the averageaverage (or (or meanmean).).



STATISTICAL INFERENCE STATISTICAL INFERENCE

PopulationPopulation

SampleSample

Statistical inferenceStatistical inference

CensusCensus

Sample surveySample survey

the set of all elements of interest in athe set of all elements of interest in a particular studyparticular study

a subset of the populationa subset of the population

the process of using data obtainedthe process of using data obtained from a sample to make estimatesfrom a sample to make estimates and test hypotheses about theand test hypotheses about the characteristics of a populationcharacteristics of a population

collecting data for a populationcollecting data for a population

collecting data for a samplecollecting data for a sample



COMPUTERS AND STATISTICAL ANALYSIS

Statistical analysis typically involves working withStatistical analysis typically involves working with large amounts of datalarge amounts of data..

Computer softwareComputer software is typically used to conduct the is typically used to conduct the analysis.analysis.



DESCRIPTIVE STATISTICS:TABULAR AND GRAPHICAL PRESENTATIONS

• Summarizing Qualitative Data• Summarizing Quantitative Data



SUMMARIZING QUALITATIVE DATA

• Frequency Distribution• Relative Frequency Distribution • Percent Frequency Distribution• Bar Graphs• Pie Charts



A A frequency distributionfrequency distribution is a tabular summary of is a tabular summary of data showing the frequency (or number) of itemsdata showing the frequency (or number) of items in each of several non-overlapping classes.in each of several non-overlapping classes.

The objective is to The objective is to provide insightsprovide insights about the data about the data that cannot be quickly obtained by looking only atthat cannot be quickly obtained by looking only at the original data.the original data.

FREQUENCY DISTRIBUTIONFREQUENCY DISTRIBUTION


Amity Business School, Noida TABULAR SUMMARY:

FREQUENCY AND PERCENT FREQUENCY

Very GoodVery Good GoodGood Above AverageAbove Average AverageAverage PoorPoor Very PoorVery Poor

22 1313 1616 77 77 55 5050

44 2626 3232 1414 1414 1010 100100

(2/50)10(2/50)1000

Water Water QualityQuality

Frequency Frequency DistributionDistribution

PercentPercentFrequencyFrequency

For example groundwater quality of 50 samples


The The relative frequencyrelative frequency of a class is the fraction or of a class is the fraction or proportion of the total number of data itemsproportion of the total number of data items belonging to the class.belonging to the class.

A A relative frequency distributionrelative frequency distribution is a tabular is a tabular summary of a set of data showing the relativesummary of a set of data showing the relative frequency for each class.frequency for each class.

RELATIVE FREQUENCY RELATIVE FREQUENCY DISTRIBUTIONDISTRIBUTION



PERCENT FREQUENCY DISTRIBUTION

The The percent frequencypercent frequency of a class is the relative of a class is the relative frequency multiplied by 100.frequency multiplied by 100.

AA percent frequency distributionpercent frequency distribution is a tabular is a tabular summary of a set of data showing the percentsummary of a set of data showing the percent frequency for each class.frequency for each class.



RELATIVE FREQUENCY AND PERCENT FREQUENCY RELATIVE FREQUENCY AND PERCENT FREQUENCY DISTRIBUTIONSDISTRIBUTIONS

Very GoodVery Good GoodGood Above AverageAbove Average AverageAverage PoorPoor Very PoorVery Poor

.04.04 .26.26 .32.32 .14.14 .14.14 .10.10

TotalTotal 1.00 1.00

0404 2626 3232 1414 1414 1010100100

RelativeRelativeFrequencyFrequency

PercentPercentFrequencyFrequencyRatingRating

.04(100) = .04(100) = 44

5/50 5/50 = .10= .10


Amity Business School, NoidaBAR GRAPH

A A bar graphbar graph is a graphical device for depicting is a graphical device for depicting qualitative data.qualitative data. On one axis (usually the horizontal axis), we specifyOn one axis (usually the horizontal axis), we specify the labels that are used for each of the classes.the labels that are used for each of the classes. A A frequencyfrequency, , relative frequencyrelative frequency, or , or percent frequencypercent frequency scale can be used for the other axis (usually thescale can be used for the other axis (usually the vertical axis).vertical axis). Using a Using a bar of fixed widthbar of fixed width drawn above each class drawn above each class label, we extend the height appropriately.label, we extend the height appropriately. The The bars are separatedbars are separated to emphasize the fact to emphasize the fact that eachthat each class is a separate category.class is a separate category.



Poor BelowAverage

Average AboveAverage

Excellent

Freq

uenc

y

Rating123456789

10 Water Quality RatingsWater Quality Ratings


Amity Business School, NoidaPIE CHART

The The pie chartpie chart is a commonly used graphical device is a commonly used graphical device for presenting relative frequency distributions forfor presenting relative frequency distributions for qualitative data.qualitative data. First draw a First draw a circlecircle; then use the relative; then use the relative frequencies to subdivide the circlefrequencies to subdivide the circle into sectors that correspond to theinto sectors that correspond to the relative frequency for each class.relative frequency for each class. Since there are 360 degrees in a circle, Since there are 360 degrees in a circle, a class with a relative frequency of .25 woulda class with a relative frequency of .25 would consume .25(360) = 90 degrees of the circle.consume .25(360) = 90 degrees of the circle.



BelowAverage 15%

Average 25%

AboveAverage 45%

Poor10%

Excellent 5%

Watrer Quality RatingsWatrer Quality Ratings



SUMMARIZING QUANTITATIVE DATA

• Frequency Distribution• Relative Frequency Distribution• Percent Frequency Distribution• Histogram• Cumulative Distributions• Ogive



TDS (ppm) for 50 samples of a groundwater TDS (ppm) for 50 samples of a groundwater of a districtof a district

91 78 93 57 75 52 99 80 97 6271 69 72 89 66 75 79 75 72 76104 74 62 68 97 105 77 65 80 10985 97 88 68 83 68 71 69 67 7462 82 98 101 79 105 79 69 62 73

91 78 93 57 75 52 99 80 97 6271 69 72 89 66 75 79 75 72 76104 74 62 68 97 105 77 65 80 10985 97 88 68 83 68 71 69 67 7462 82 98 101 79 105 79 69 62 73



FREQUENCY DISTRIBUTION• Guidelines for Selecting Width of Classes

Largest Data Value Smallest Data ValueNumber of Classes

• Use classes of equal width.Use classes of equal width.

• Approximate Class Width =Approximate Class Width =



50-5950-59 60-69 60-69 70-7970-79 80-8980-89 90-9990-99 100-109100-109

22 1313 1616 77 77 55Total 50Total 50

TDS(ppm)TDS(ppm) FrequencyFrequency

Approximate Class Width = (109 - 52)/6 = 9.5 Approximate Class Width = (109 - 52)/6 = 9.5 1010



RELATIVE FREQUENCY ANDPERCENT FREQUENCY DISTRIBUTIONS

50-5950-59 60-69 60-69 70-7970-79 80-8980-89 90-9990-99 100-109100-109

TDSTDS(ppm(ppm)) .04.04 .26.26 .32.32 .14.14 .14.14 .10.10Total 1.00 Total 1.00

RelativeRelativeFrequencyFrequency

44 2626 3232 1414 1414 1010 100100

PercentPercent FrequencyFrequency

2/502/50 .04(10.04(100)0)



• Only 4% of the 50 samples have TDS in the 50-59 class.Only 4% of the 50 samples have TDS in the 50-59 class.

• The greatest percentage (32% or almost one-third)The greatest percentage (32% or almost one-third) of the samples have TDS in the 70-79 class.of the samples have TDS in the 70-79 class.

• 30% of the samples have TDS under 70ppm.30% of the samples have TDS under 70ppm.

• 10% of the samples have TDS of 100ppm or more.10% of the samples have TDS of 100ppm or more.

Insights Gained from the Percent Frequency Insights Gained from the Percent Frequency DistributionDistribution

RELATIVE FREQUENCY ANDRELATIVE FREQUENCY ANDPERCENT FREQUENCY DISTRIBUTIONSPERCENT FREQUENCY DISTRIBUTIONS



DOT PLOTDOT PLOT

One of the simplest graphical summaries of One of the simplest graphical summaries of data is a data is a dot plotdot plot..

A horizontal axis shows the range of data A horizontal axis shows the range of data values.values.

Then each data value is represented by a dot Then each data value is represented by a dot placed above the axis.placed above the axis.



5050 6060 7070 8080 9090 100100 110110

TDS (ppm)

Groundwater samples (TDS)Groundwater samples (TDS)



HISTOGRAM Another common graphical presentation ofAnother common graphical presentation of quantitative data is a quantitative data is a histogramhistogram.. The variable of interest is placed on the horizontalThe variable of interest is placed on the horizontal axis.axis. A rectangle is drawn above each class interval withA rectangle is drawn above each class interval with its height corresponding to the interval’s its height corresponding to the interval’s frequencyfrequency,, relative frequencyrelative frequency, or , or percent frequencypercent frequency.. Unlike a bar graph, a histogram has Unlike a bar graph, a histogram has no naturalno natural separation between rectanglesseparation between rectangles of adjacent classes. of adjacent classes.


Amity Business School, Noida GRAPHICAL SUMMARY: HISTOGRAM

2468

1012141618

TDS(ppm)

Freq

uenc

y

5059 6069 7079 8089 9099 100-110

TDS TDS


• SYMMETRIC HISTOGRAM– Left tail is the mirror image of the right tailRe

lativ

e Fr

eque

ncy

.05

.10

.15

.20

.25

.30

.35

0



• MODERATELY SKEWED LEFT– A longer tail to the left

Rela

tive

Freq

uenc

y

.05

.10

.15

.20

.25

.30

.35

0



• MODERATELY RIGHT SKEWED– A longer tail to the right

Rela

tive

Freq

uenc

y

.05

.10

.15

.20

.25

.30

.35

0



• HIGHLY SKEWED RIGHT– A very long tail to the rightRe

lativ

e Fr

eque

ncy

.05

.10

.15

.20

.25

.30

.35

0



Cumulative frequency distributionCumulative frequency distribution shows the shows the numbernumber of items with values less than or equal to the of items with values less than or equal to the upper limit of each class..upper limit of each class..

Cumulative relative frequency distributionCumulative relative frequency distribution – shows – shows the the proportionproportion of items with values less than or of items with values less than or equal to the upper limit of each class.equal to the upper limit of each class.

CUMULATIVE DISTRIBUTIONSCUMULATIVE DISTRIBUTIONS

Cumulative percent frequency distributionCumulative percent frequency distribution – shows – shows the the percentagepercentage of items with values less than or of items with values less than or equal to the upper limit of each class.equal to the upper limit of each class.



<< 59 59 << 69 69 << 79 79 << 89 89 << 99 99<< 109 109

TDS (ppm)TDS (ppm) CumulativeCumulativeFrequencyFrequency

CumulativeCumulativeRelativeRelativeFrequencyFrequency

CumulativeCumulativePercentPercent FrequencyFrequency

22 1515 3131 3838 4545 50 50

.04.04 .30.30 .62.62 .76.76 .90.90 1.00 1.00

44 3030 6262 7676 9090 100 100

2 + 2 + 1313

15/5015/50 .30(10.30(100)0)



OGIVEOGIVE

An An ogiveogive is a graph of a cumulative is a graph of a cumulative distribution.distribution. The data values are shown on the horizontal The data values are shown on the horizontal axis.axis. Shown on the vertical axis are the:Shown on the vertical axis are the:• cumulative frequencies, orcumulative frequencies, or• cumulative relative frequencies, orcumulative relative frequencies, or• cumulative percent frequenciescumulative percent frequencies

The frequency (one of the above) of each class The frequency (one of the above) of each class is plotted as a point.is plotted as a point.

The plotted points are connected by straight The plotted points are connected by straight lines.lines.



• Because the class limits for the groundwater Because the class limits for the groundwater TDS data are 50-59, 60-69, and so on, there TDS data are 50-59, 60-69, and so on, there appear to be one-unit gaps from 59 to 60, appear to be one-unit gaps from 59 to 60, 69 to 70, and so on.69 to 70, and so on.• These gaps are eliminated by plotting points These gaps are eliminated by plotting points halfway between the class limits.halfway between the class limits.

• Thus, 59.5 is used for the 50-59 class, 69.5 Thus, 59.5 is used for the 50-59 class, 69.5 is used for the 60-69 class, and so on.is used for the 60-69 class, and so on.



TDSTDS20

40

60

80

100

Cum

ulat

ive

Perc

ent F

requ

ency

Cum

ulat

ive

Perc

ent F

requ

ency

50 60 70 80 90 100 11050 60 70 80 90 100 110

(89.5, 76)(89.5, 76)

Ogive with Cumulative Percent Ogive with Cumulative Percent Frequencies Frequencies



THANK YOU

54


modelling and statistical analysis

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