Elementary statistics for Food Indusrty

Elementary statistics 1

ELEMENTARY STATISTICS

FORFOOD

INDUSTRY

By

Atcharaporn Khoomtong


CONTENT Introduction Statistical methodology Step of scientific research Important parametric tests Important nonparametric tests Example using Excel program Using Excel for Statistics in Gateway

Cases – Office 2007


INTRODUCTIONMost people become familiar with probability

and statistics through radios, television,newspapers and magazines.For example,the following statements were found in newspapers.

Eating 10 grams(g) of fiber a day reduce the risk

of heart attack by 14% Thirty minutes (of exercise) two or three times

each week can raise HDLs 10 to 15%


Statistics is used to analyze the results of surveys and as a tool in scientific research to make decisions based on controlled experiments.

Other uses of statistics include operations research, quality control, estimation and prediction.

INTRODUCTION

What’s it?

Flower


STATISTICS AND STATISTICAL METHODOLOGY

as the basis of data analysis are concerned with two basic types of problems

(1) summarizing, describing, and exploring the data

(2) using sampled data to infer the nature of the process which produced the dataThis problems is covered by inferential statistics.

This problems is covered by descriptive statistics


DESCRIPTIVE STATISTICS Statistics plays an important role in the

description of mass phenomena. Organized and summarized for clear

presentation for ease of communications. Data may come from studies of

populations or samples It offers methods to summarize a

collection of data. These methods may be numerical or graphical, both of which have their own advantages and disadvantages.


Inferential statistics is used to draw conclusions about a data set.

Usually this means drawing inferences about a population from a sample either by estimating some relationships or by testing some hypothesis.

INFERENTIAL STATISTICS

A Population is the set of all possible states of a random variable. The size of the population may be either infinite or finite.

A Sample is a subset of the population; its size is always finite.

EXAMPLES OF DESCRIPTIVE AND INFERENTIAL STATISTICS

Descriptive Statistics

Graphical Arrange data in tables Bar graphs and pie

charts Numerical

Percentages Averages Range

Relationships Correlation coefficient Regression analysis

Inferential Statistics Confidence interval Compare means of

two samples t Test F -Test

Compare means from three samples Pre/post (LSD,DMRT) ANOVA = analysis of

variance F -Test


Another important aspect of data analysis is the Data, which can be of two different types:

qualitative data ex. Sex, color, smell, taste etc. quantitative data ex. Height, weight, percentage etc.

Qualitative data does not contain quantitative information.

Qualitative data can be classified into categories.


SCALESType of Scale Possible

StatementsAllowed Operators

Examples

nominal scale identity, countable =, ≠ colors, phone numbers, feelings

ordinal scale identity, less than/greater than relations, countable

=, ≠, <, >

soccer league table, military ranks, energy efficiency classes

interval scale identity, less than/greater than relations, equality of differences

=, ≠ , <, >, +, -

dates (years), temperature in Celsius, IQ scale

ratio scale identity, less than/greater than relations, equality of differences, equality of ratios, zero point

=, ≠ , <, >, +, -, *, /

velocities, lengths, temperatur in Kelvin, age

STEP OF SCIENTIFIC RESEARCH

Collecting the necessary

facts Analyzing the facts

Making decisions

Carrying out decisions

Assessing the results

Descriptive Statistics

Inference Statistics

DESCRIPTIVE STATS VOCABULARY Mode =The most frequent value Median =The value of the middle point of the ordered

measurements Mean =The average (balancing point in the

distribution) Variance= The average of the squared deviations of

all

the population measurements from the

population mean Standard deviation =The square root of the variance

DESCRIPTIVE VS. INFERENTIAL FORMULAS

2

2

Descriptive FormulaDescriptive Formula

1

2

2

S

Inferential FormulaInferential Formula

Called the “unbiased estimator of the

population value”

EXAMPLE: VARIANCE AND STANDARD DEVIATION

Population of profit margins for five companies:8%, 10%, 15%, 12%, 5%

6115

58

5

25425045

52502

5

105101210151010108

22222

222222

.

%40636112 ..

%105

50

5

51215108

INFERENTIAL STATS VOCABULARY

Hypothesis = a assumption or some supposition

to be proved or disproved.

“the automobile A is performing as well as

automobile B.”


Null hypothesis (H0 ) =expresses no difference

Alternative hypothesis (H1 )


H0: = 0

Often said “H naught” Or any number

Later…….H0: 1 = 2

H0: = 0; Null Hypothesis

HA: = 0; Alternative Hypothesis


TYPE OF ERRORType I error (α) :

reject H0 | H0 true

Type II error (β) : Accept H0 | H1 true



Calculated F value is greater than the critical F values

Significant >>>reject H0

Calculated F value is lower than the critical F values

Non Significant >>>accept H0

UNKNOWN TRUTH AND THE DATA

α = significance level1- β = power

Truth Data

H0 Correct HA Correct

Decide H0

“fail to reject H0”

1- αTrue Negative

βFalse Negative

Decide HA

“reject H0”

αFalse Positive

1- βTrue Positive


IMPORTANT PARAMETRIC TESTS

Z - test

T – test

F – test

Z - test IMP

OR

TA

NT P

AR

AM

ETR

IC

TES

TSis based on the normal probability

distribution and is used for judging the significance of several statistical measures, particularly the mean. (n>30)

z-test is generally used for comparing the mean of a sample to some hypothesized mean for the population in case of large sample


T – testis based on t-distribution and is considered an appropriate test for judging the significance of a sample mean or for judging the significance of difference between the means of two samples in case of small sample(s) when population variance is not known (in which case we use variance of the sample as an estimate of the population variance).

t-test applies only in case of small sample(s) when population variance is unknown.

IMP

OR

TA

NT P

AR

AM

ETR

IC T

ES

TS

Unknown varianceUnder H0

Critical values: statistics books or computert-distribution approximately normal for degrees of freedom (df) >30

0( 1)~

/n

Xt

s n


F – testis based on F-distribution and is used to compare the variance of the two-independent samples. This test is also used in the context of analysis of variance (ANOVA) for judging the significance of more than two sample means at one and the same time.

Test statistic, F, is calculated and compared with its probable value (to be seen in the F-ratio tables for different degrees of freedom for greater and smaller variances at specified level of significance) for accepting or rejecting the null hypothesis.

IMP

OR

TA

NT P

AR

AM

ETR

IC T

ES

TS

Anova tables:

for a 1-way anova with N observations and T treatments.

Source df SS MS Ftreatment (T-1) SStrt =SStrt/(T-1) MStrt/MSerr

error…………by subtraction Sserr =SSerr/dferr

Total (N-1)

Finally, you (or the PC) consult tables or otherwise obtain a probability of obtaining this F value given df for treatment and error.

HOW TO DO AN ANOVA BY HAND:

1: Calculate N, Σx, Σx2 for the whole dataset.2: Find the Correction factor

CF = (Σx * Σx) /N3: Find the total Sum of Squares for the data

= Σ(xi2) – CF

4: add up the totals for each treatment in turn (Xt.), then calculate Treatment Sum of Squares

SStrt = Σt(Xt.*Xt.)/r - CF

where Xt. = sum of all values within treatment t, and r is the number of observations that went into that total.

3: Draw up ANOVA table, getting error terms by subtraction.


EXPERIMENTAL DESIGN

Complete Randomize Design (CRD)

Randomize Complete Block Design (RBD)

Latin Square (LQ)TreatmentsReplicationDegree of

freedom (df)

@LSD : Least Significant Difference

@DMRT:Duncan’s New Multiple Range Test

Requir

e


LINEAR VS.NONLINEAR MODELS

Most people have difficulties in determining whether a model is linear or non-linear.

Before discussing the issues of linear vs. non-linear systems, let's have a short look at some examples, displaying several types of discrimination lines between two classes:

linear

Non-linear

IMP

OR

TA

NT P

AR

AM

ETR

IC T

ES

TS


HAVE YOU ALREADY GUESSED THE DIFFERENCE BETWEEN LINEAR AND NON-

LINEAR MODELS ?

Here's the answer: linear models are linear in the parameters which have to be estimated, but not necessarily in the independent variables.

This explains why the middle of the three figures above shows a linear discrimination line between the two classes, although the line is not linear in the sense of a straight line.


REGRESSION MODEL

When calculating a regression model, we are interested in a measure of the usefulness of the model.

There are several ways to do this, one of them being the coefficient of determination (also sometimes called goodness of fit).

The concept behind this coefficient is to calculate the reduction of the error of prediction when the information provided by the x values is included in the calculation.


Thus the coefficient of determination specifies the amount of sample variation in y explained by x.

For simple linear regression the coefficient of determination is simply the square of the correlation coefficient between Y and X .

CO

EFFIC

IEN

T O

F D

ETER

MIN

ATIO

N

-1 +10

Strong negativeLinear relationship

Strong positiveLinear relationship

No Linear relationship


THE CORRELATION COEFFICIENT R

also called Pearson's product moment correlation after Karl Pearson is calculated by

The correlation coefficient may take any value between -1.0 and +1.0.

Assumptions: linear relationship between x and y continuous random variables both variables must be normally distributed x and y must be independent of each other


IMPORTANT NONPARAMETRIC TESTS

c2 test

c2 testis based on chi-square distribution and as a parametric test is used for comparing a sample variance to a theoretical population variance.

IMP

OR

TA

NT P

AR

AM

ETR

IC T

ES

TS

where = variance of the sample;

= variance of the population; (n – 1) = degrees of freedom, n being the number of items in the sample.


EXAMPLE USING EXCEL PROGRAM


DETERMINING CONFIDENCELIMITS FOR A POPULATION MEANUSING T-DISTRIBUTION

In quality control, there are situations when we need to know whether a sample mean lies within the confidence limits of the entire population. This can be accomplished by using t-distribution to determine confidence limits for a population mean using a selected probability.

EXAMPLE

I

We will use Excel function TINV( ) to determine the t-distribution.


PROBLEM STATEMENT: Ten cans of sliced pineapple were

removed at random from a population of 1000 cans. The drained weight of the contents were measured as 410.5, 411.4, 410.4, 412.6, 411.9, 411.5,412.5, 411.4, 411.5, 410.1 g. Determine the 95% confidence limits for the entire population.


APPROACH: We will first calculate the average of the

ten data values using the AVERAGE() function. Next we will determine the standard deviation of the sample mean using STDEV() function. Then we will use the following expression to estimate the lower and upper limits of population mean


PROGRAMMING THE WORKSHEET:

Discussion:The results show that the 95% confidence lower and upper limits for the population mean are 410.78 and 411.98, respectively.


STATISTICAL DESCRIPTORS OF A POPULATION ESTIMATED FROM SELECTED DATA OBTAINED FOR A SAMPLE

When a sample is taken from a large population and analyzed for selected DATA, statistical analysis is helpful in obtaining estimates for the total population from which the sample was obtained. In this worksheet.

EXAMPLE

IIWe will use Excel's built-in data analysis techniques to determine various statistical descriptors for the sample and the population.


PROBLEM STATEMENT:

A sample of 10 breads is obtained from a conveyor belt exiting a baking oven. The breads are analyzed for color by comparing them with a standard color chart. The values

recorded, in customized color units, are as follows: 34, 33, 36,37, 31, 32, 38, 33, 34, and 35. Estimate the mean, variance,

and standard deviation of the population.

Case study : Color Data


APPROACH: We will use the Data Analysis capability

of Excel in determining the descriptive statistics for the given data. First, you should make sure that Data Analysis... is available

under the menu command Tools. If it is not available, then see Next slide for details on how to add this analysis package.

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TO SET UP STATISTICAL PACKAGE

Click Microsoft Office Button , and Then Click Excel Options

Click Add-ins. In Manage Box, Select Excel Add-ins

Click Go In the Add-Ins Available Box, Select Analysis

ToolPak Check Box and Click OK. (If ToolPak Is Not Listed, Click Browse to Locate It.)



Step 1 Open a new worksheet expanded to full size.Step 2 In cells A2 :A 11, type the text labels and data values


Step 3 Choose the menu items Data, Data Analysis .... A dialog box will open as shown.Step 4 Double click on Descriptive Statistics.


Step 5 In the edit box for Input Range:, type the range ofcells as SA$2:$A$11.Step 6 Select the radio button Columns.Step 7 In output range type A13. Click OK.Step 8 Excel will calculate the descriptive statistics anddisplay results in cells A13:B28

@The results indicate that the sample mean is 34.3. @The standard deviation forthe population is 2.214, and @the sample variance of thepopulation is 4.9

STATISTICAL TESTING: T-TEST

t (difference between samples) / (variability)

Excel will automatically calculate t-values to compare:Means of two datasets with equal variancesMeans of two datasets with unequal variancesTwo sets of paired data

abs(t-score) < abs(t-critical): accept H0

Insufficient evidence to prove that observed differences reflect real, significant differences

47


PROBLEM STATEMENT: A researcher wishes to test whether heavy

metal in soil have different mean after war threat versus before war threat. The heavy metal in soil is that mean after war threat will exceed mean before war threat

Use Excel to help test the hypothesis for the difference in population means.

EXAMPLE

III


Step 1 Open a new worksheet expanded to full size.Step 2 In cells B5 :C19, type the text labels and data values

The null and hypothesis to be test are:

0.0:

0.0:

21

21

A

o

H

H


Step 3 Choose the menu items Tools, Data Analysis .... A dialog box will open as shown.Step 4 Double click on t-Test two-sample assuring equal variances.



Change this if you want to know whether the means of the two samples differ by at least some specified amount.

p value for Two-tail test is .007 which is less than .05 so we reject the null hypothesis.

p value for one tailed test is .003 which is less than .05 so we reject the null hypothesis.

t > tcritical(two-tail), so the mean of sample #1 is significantly different from the mean of sample #2.

t > tcritical(one-tail), so the mean of sample #1 is significantly larger than the mean of sample #2.


ANALYSIS OF VARIANCE: ONE FACTOR, COMPLETELY RANDOMIZED DESIGN

In hypothesis testing, it is sometimes not possible to use the same judges for testing different treatments. Although, it would be desirable to use the same judges to evaluate samples obtained from different treatments.

In such cases, we have a completely randomized design. Using single-factor ANOVAWe can test to see whether the treatments had any influence on the

judges scores; in other words, does the mean of each treatment differ?

EXAMPLE

IV


PROBLEM STATEMENT:

Consider a weight of oranges from three different suppliers A, B, and C .Five oranges was random sampling and weighted. The following weights were obtained:

Case study : Weight of oranges Data

A B C

150 148 146

151 150 148

152 152 150

153 154 152

154 156 154


PROBLEM STATEMENT (CON’T)

For each treatment, 5 samples were weighted by 5 times. Therefore, the design was completely randomized. Calculate the F value to determine whether the means of three treatments are significantly different.


APPROACH:

We will use a single factor analysis of variance available in Excel. We will determine the F value at probability of 0.95 .

These computations will allow us to determine if the means between the three different treatments are significantly different.

First make sure that the Data Analysis... Command is available under menu item Data.


PROGRAMMING THE WORKSHEET:Step 1 Open a new worksheet expanded to full size.Step 2 In cells A4 :C8, type the text labels and data values


Step 3 Choose the menu items Data, Data Analysis .... A dialog box will open as shown.Step 4 Double click on Anova Single Factor.


The results show that the F value is 0.889. The critical Fvalues are At the 5% level F = 3.885

This indicates that for the example problem the F value is lower than the value at the 5% level but not at the 5% level. Thus, we cansay that no significant difference in their mean scores(P<0.05).


ANALYSIS OF VARIANCE FOR ATWO-FACTOR DESIGN WITHOUTREPLICATION

When we are interested in evaluating samples for sensory characteristics using same judges with samples obtained from multiple treatments, analysis of variance for a two-factor design without replication is useful.

This analysis helps in determining if there are significant differences among the various treatments as well as if an significant differences exist among the judges themselves.

EXAMPLE

V


PROBLEM STATEMENT: Three types of ice cream were evaluated

by 11 judges. The judges assigned the following scores.

Judge Ice Cream A Ice Cream B

Ice Cream C

A 16 14 15

B 17 15 17

C 16 16 16

D 18 14 16

E 16 14 14

F 17 16 17

G 18 14 15

H 16 15 16

I 17 14 14

J 18 13 16

K 17 15 15


APPROACH:

We will use the built-in analysis pack available in the Excel command called Data Analysis ....

Three sets of results will be obtained for the 5% level



Step 1 Open a new worksheet expanded to full size.Step 2. In cell A3 :D 13, type the text labels and data values,


Step 3 Choose the menu items Data, Data Analysis ....A dialog box will open.Step 4 Double click on Anova: Two-Factor WithoutReplication. A new dialog box will open.Step 5 Type entries in edit boxes as shown.Step 6. The results will be displayed in cells


The difference among ice cream types is determined by examining the F values. The F value is calculated as 19.73. This value is greater than 3.49 for the 5% level

For judges, the calculated F value is 1.36. This value is lower than the critical F values of 2.35 at the 5 % level


DISCUSSION The difference among ice cream types is

determined by examining the F values. The F value is calculated as 19.73. This value is greater than 3.49 for the 5% level,

The ice cream types are significantly different at p<0.001.

For judges, the calculated F value is 1.36. This value is lower than the critical F values of 2.35 at the 5 % level.

The judges showed no significant difference in their mean scores.


USE OF LINEAR REGRESSION INANALYZING DATA

Simple regression analysis involves determining the statistical relationship between two variables. One of the uses of such analysis is in predicting one variable on the basis of the other.

We will use the regression analysis available inthe Add-in package in Excel to determine linear regressionbetween two variables.

EXAMPLE

VI


PROBLEM STATEMENT:

flavor with storage time in a frozen vegetable. Sensory scores obtained at 0, 1, 2, 3, 4 and 6 month times were 1.5, 2, 2, 3,

2.5, and 3.5, respectively. Assuming that these data can be linearly correlated, determine the regression coefficient and

predict the off-flavor score at 5 months of storage.

Case study : Sensory scores Data


APPROACH: We will use the package Regression

available as an Add-in item in Excel. We will use this package to obtain required statistical relationships. We assume that a linear relationship exists between the off-flavor score and time (in months) with the equation

y= mx+b,

where y is off-flavor score, x is time in months, m is slope andb is intercept.



Step 1 Open a new worksheet expanded to full size.Step 2 In cells A4 :B9, enter the text labels and data values


Step 3 Choose the menu items Data, Data Analysis .... A dialog box will open.Step 4 Double click on Regression.Step 5 A new dialog box will open. Enter the range of cells for Y and X as shown. Check boxes for Residuals and Line Fit Plots. Click OK.

Probability of getting this value of F by randomly sampling from a normally distributed population. Low value means model (rather than random variability) explains most variation in data.

Ratio of variability explained by model to leftover variability. High number means model explains most variation in data.

~99% of the variation in y is explained by variation in x. The remainder may be random error, or may be explained by some factor other than x.

Confidence limits on slope and intercept.

Probability of getting a slope or intercept this much different from zero by randomly sampling from a normally-distributed population.

y =

0.

31 x

+

1.58

The results will

be displayed


DISCUSSION

The r 2 value is calculated as 0.85, the standard error is 0.318.The intercept is 1.5786 and the slope is 0.3143.

The linear equation is y = 0.31x + 1.58 . The residual output gives the predicted values for the off-flavor score at different time intervals. These data are also shown in the chart.

The predicted and calculated values are shown. The predicted value at 5 months of storage duration is calculated as 3.13.


A PLOT OF LINEAR REGRESSION OF DATA

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USING EXCEL FOR STATISTICS IN

GATEWAY CASES – OFFICE 2007

76

CONCEPTS COVERED

Statistics

- Descriptive Statistics

- Histograms

- Hypothesis Testing

- Scatter Plots

- Regression Analysis

77

TO SET UP STATISTICAL PACKAGE

Click Microsoft Office Button , and Then Click Excel Options

Click Add-ins. In Manage Box, Select Excel Add-ins

Click Go In the Add-Ins Available Box, Select Analysis

ToolPak Check Box and Click OK. (If ToolPak Is Not Listed, Click Browse to Locate It.)

78

USING EXCEL:DESCRIPTIVE STATISTICS

Click Data/Data Analysis (Far Right) /Descriptive Statistics & OK.

Put Checkmarks on Summary Statistics, 95% or 99% Confidence Interval, & Labels in First Row

Boxes. Move Cursor to Input Range Window, Highlight Data to Analyze including Labels, & Click OK. Your Data will Appear on New Worksheet. Widen Columns by Clicking Home/Format/AutoFit

Column Width.

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USING EXCEL:CONSTRUCTING HISTOGRAMS

Click Data/Data Analysis/Histogram & OK. Put Checkmarks on Chart Output & New Worksheet

Boxes. Move Cursor to Input Range Window, Highlight Data

Going into Histogram. Move Cursor to Input Bin Range, Highlight Data Showing Upper Value of Each Bin & Click OK. Histogram will be on New Worksheet. You May Lengthen it by Clicking Blank Space in Window, Moving

Cursor to Window Bottom Line & Holding Down Mouse Button as You Pull Down Window.

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USING EXCEL:HYPOTHESIS TESTING

Go to Sheet One. Click Data/Data Analysis/ and the Appropriate

Statistical Test. Then Click OK. On New Window Check Labels Box and Put

Cursor on Variable 1 Range. Highlight Variable 1 Data Including Label. Put Cursor on Variable 2 Range & Highlight

Variable 2 Data (Including Label). Then Click OK. Click Home/Format/AutoFit/Column Width

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USING EXCEL:SCATTER PLOTS

Go to Sheet One. Highlight Data (Be Sure X Values are in

Left Column and Y Values are in Right Column).

Click Insert/Scatter. Pull down menu and click Upper Left Icon.

Click a Datum Point on Chart with Right Mouse Key, Add Trendline, & Click Linear.

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USING EXCEL:REGRESSION ANALYSIS

Go to Sheet One. Click Data/Data Analysis (On Far Right)

/Regression & Click OK. On New Window Check Labels Box and Put

Cursor on X Range. Highlight X Data Including Label. Put Cursor on Y Range & Highlight Y Data

(Including Label), Then Click OK. Click Home/Format/AutoFit Column Width.


Education

Elementary statistics for Food Indusrty