class interval class boundaries class mark, x i frequency, f i relative freq’y. %

Probability and StatisticsCourse Requirements1. Quizzes – 25%2. First Long Exam – 25%3. Second Long Exam – 25%4. Third Long Exam – 25%

5. Total – 100%

Passing – 60%

Probability and StatisticsStatisticsA branch of mathematics that deals with the

collection, organization and analysis of numerical data and with such problems as experiment design and decision making.

3 Important features of Statistics:

1. Data gathering

2. Data analysis3. Making decision

Probability and StatisticsDefinition of terms

1. Raw data Data collected in original form

2. Variable Characteristic or attribute that can assume

different values3. Population

All subjects possessing a common characteristic that is being studied


4. Sample A subgroup or subset of a population

5. Parameter Characteristic or measure obtained from a

population6. Qualitative variables

Variables which assume non-numerical values


7. Quantitative variables variables which assume numerical values

8. Discrete variables Variables which assume finite or countable

number of possible values, usually obtained by counting

9. Continuous variables Variables which assume infinite number of

possible values, usually obtained by measurement

Probability and Statistics Everyone involved in the experiment must have a

clear idea about what is to be studied, how the data is to be collected and at least a qualitative understanding as to how these data are to be analyzed.

Guidelines for designing experiments:1. Statement of the problem / recognition of the

problem Develop all the ideas about the objectives of

the experiment

Probability and StatisticsGuidelines for designing experiments:2. Choice of factors and levels

Choose the factors to be varied in the experiment

Choose the ranges over which these factors will be varied

Identify the specific levels at which runs will be made

Probability and StatisticsGuidelines for designing experiments:3. Selection of the response variable

The experimenter should be certain that this variable really provides useful information about the process under study

4. Choice of experimental design Involves the consideration of sample size

(number of replicates/trials), the selection of a suitable run order for the experimental trials, and the determination of whether or not blocking or other randomization restrictions involved.

Probability and StatisticsGuidelines for designing experiments:5. Performing the experiment

Monitor the process carefully to ensure that everything is being done according to plan

6. Data analysis Analyzing the data collected during the

experiment by statistical methods6. Conclusions

Making decision based on the statistical results

Probability and StatisticsMethods of Sampling1. Random sampling

sampling in which the data is collected using chance methods or random numbers.

2. Systematic sampling Sampling in which the data is collected by

selecting every kth object3. Stratified sampling

Sampling in which the population is divided into groups (strata) according to some characteristic. Each strata is then sampled either random or systematic

Probability and StatisticsMethods of Sampling4. Cluster sampling

sampling in which the population is divided into groups (usually geographically). Some of these groups are randomly selected, and then all of the elements in those groups are selected.

Probability and StatisticsMethods of Summarizing/Characterizing Data 1. Tabular Methods

a. Frequency Distribution

b. Cumulative Frequencyc. Stem and Leaf Table

2. Graphical Methodsa. Frequency Histogram

b. Frequency Polygonc. Ogived. Pie chart

Probability and StatisticsMethods of Summarizing/Characterizing Data 3. Numerical Methods

a. Measures of Central Tendencies

b. Measures of Dispersion

Mean/Average, Median, Mode

Range, Variance, Standard Deviationc. Measures of Shape

Skewness, Kurtosisd. Measures of Data Locations

Percentiles, Deciles, Quartiles

Probability and StatisticsTabular Methods1. Frequency Distribution

The organization of raw data in tabular form with classes and frequencies

Steps in Constructing a Frequency Distribution Table:1. Determine the number of class intervals, k, needed

to summarize the data:

No. of samplesNo. of class intervals

Probability and StatisticsTabular MethodsSteps in Constructing a Frequency Distribution Table:2. Find the range of observations

Minimum valueRangeMaximum value

Probability and StatisticsTabular MethodsSteps in Constructing a Frequency Distribution Table:3. Determine the width of the class intervals

No. of class intervals

Class width

Range

Probability and StatisticsTabular MethodsSteps in Constructing a Frequency Distribution Table:4. Form the frequency table

Class Interval

Class Boundaries

Class Mark,xi

Frequency,fi

Relative Freq’y.

%

Class interval Separates one class in a grouped frequency from

the other The interval could actually appear in the raw data

and it begins with the lowest value


Class Interval

Class Boundaries

Class Mark,xi

Frequency,fi

Relative Freq’y.

%

Class boundary Separates one class in a grouped frequency from

the other It has one more decimal place than the raw data

and therefore it does not appear in the data


Class Interval

Class Boundaries

Class Mark,xi

Frequency,fi

Relative Freq’y.

%

Class boundary


Class Interval

Class Boundaries

Class Mark,xi

Frequency,fi

Relative Freq’y.

%

Class Mark (Midpoint), xi The number in the middle of the class


Class Interval

Class Boundaries

Class Mark,xi

Frequency,fi

Relative Freq’y.

%

Frequency, fi The number of times a certain value or class of

values occurs


Class Interval

Class Boundaries

Class Mark,xi

Frequency,fi

Relative Freq’y.

%

Relative Frequency, % Frequency divided by the total number of data This gives the percent of values falling in that class

Probability and StatisticsTabular MethodsSteps in Constructing a Frequency Distribution Table:Illustration: the nicotine contents, in milligrams, for 40 cigarettes of a certain brand were recorded as follows:

1.09 1.92 2.31 1.79 2.281.74 1.47 1.97 0.85 1.241.58 2.03 1.70 2.17 2.552.11 1.86 1.90 1.68 1.511.64 0.72 1.69 1.85 1.821.79 2.46 1.88 2.08 1.671.37 1.93 1.40 1.64 2.091.75 1.63 2.37 1.75 1.69


1.09 1.92 2.31 1.79 2.281.74 1.47 1.97 0.85 1.241.58 2.03 1.70 2.17 2.552.11 1.86 1.90 1.68 1.511.64 0.72 1.69 1.85 1.821.79 2.46 1.88 2.08 1.671.37 1.93 1.40 1.64 2.091.75 1.63 2.37 1.75 1.69

Class Interval

0.72 – 1.02

1.03 – 1.33

1.34 – 1.64

1.65 – 1.95

1.96 – 2.26

2.27 – 2.57


Class Interval Class Boundaries Class Mark,

xi

0.72 – 1.02 0.715-1.025 0.871.03 – 1.33 1.025-1.335 1.181.34 – 1.64 1.335-1.645 1.491.65 – 1.95 1.645-1.955 1.801.96 – 2.26 1.955-2.265 2.112.27 – 2.57 2.265-2.575 2.42


1.09 1.92 2.31 1.79 2.281.74 1.47 1.97 0.85 1.241.58 2.03 1.70 2.17 2.552.11 1.86 1.90 1.68 1.511.64 0.72 1.69 1.85 1.821.79 2.46 1.88 2.08 1.671.37 1.93 1.40 1.64 2.091.75 1.63 2.37 1.75 1.69

Class Boundaries

Frequency,fi

0.715-1.025 21.025-1.335 21.335-1.645 81.645-1.955 171.955-2.265 62.265-2.575 5


Class Interval Class Boundaries

Class Mark,

xi

Frequency,fi

Relative Freq’y.

%

0.72 – 1.02 0.715-1.025 0.87 2 5.001.03 – 1.33 1.025-1.335 1.18 2 5.001.34 – 1.64 1.335-1.645 1.49 8 20.001.65 – 1.95 1.645-1.955 1.80 17 42.501.96 – 2.26 1.955-2.265 2.11 6 15.002.27 – 2.57 2.265-2.575 2.42 5 12.50

Probability and StatisticsTabular MethodsCumulative Frequency Distribution Table:Cumulative Frequency, cfi Gives the running total of the frequencies The number of observations in the sample whose

values are less than or equal to the upper boundary of the class interval

Relative Cumulative Frequency (cfi / total number of samples) * 100 Percent of the values which are less than the upper

boundary

Probability and StatisticsTabular MethodsCumulative Frequency Distribution Table:

Class Interval

Class Boundaries

Class Mark,

xi

Freq’y,fi

Cumulative Frequency,

cfi

Relative Cum.

Freq’y.%

0.72 – 1.02 0.715-1.025 0.87 2 2 5.00

1.03 – 1.33 1.025-1.335 1.18 2 4 10.00

1.34 – 1.64 1.335-1.645 1.49 8 12 30.00

1.65 – 1.95 1.645-1.955 1.80 17 29 72.50

1.96 – 2.26 1.955-2.265 2.11 6 35 87.50

2.27 – 2.57 2.265-2.575 2.42 5 40 100.00

Probability and StatisticsGraphical MethodsFrequency Histogram A graph which displays the data by using vertical

bars of various heights to represent frequencies The horizontal axis can either be class intervals,

class boundaries, or class marks

Probability and StatisticsGraphical MethodsFrequency Histogram

0.87 1.18 1.49 1.8 2.11 2.420

2

4

6

8

10

12

14

16

18

Class mark

freq

uenc

y

Probability and StatisticsGraphical MethodsFrequency Polygon

Class mark

freq

uenc

y

A line graph between frequency and class mark

0.87 1.18 1.49 1.8 2.11 2.420

2

4

6

8

10

12

14

16

18

Probability and StatisticsGraphical MethodsOgive

Upper class boundary

Rela

tive

cum

ulati

ve fr

eque

ncy

A frequency polygon of relative cumulative frequency against upper class boundaries

1.025 1.335 1.645 1.955 2.265 2.5750

20

40

60

80

100

120

Probability and StatisticsGraphical MethodsPie chart The degree of slice is based on the relative

frequency

552042.51512.5

Probability and StatisticsNumerical MethodsMeasures of Central Tendencies1. Mean / Average

The sum of the product of class mark and the corresponding frequency divided by the total number of samples

Probability and StatisticsNumerical MethodsMeasures of Central Tendencies2. Median

The value that will divide the samples into two equal halves when the samples are arranged from lowest to highest

Total frequencies of all class intervals before the median class

Frequency of the median class

Lower class boundary of the median class

Probability and StatisticsNumerical MethodsMeasures of Central Tendencies3. Mode

The most frequent number

Frequency difference of the modal class and the succeeding class

Frequency difference of the modal class and the preceeding class

Lower class boundary of the modal class

Probability and StatisticsNumerical MethodsMeasures of Variability / Dispersion1. Range

Measures how the samples are clustered. It is the difference between the highest and the

lowest values of the raw data

Minimum valueRangeMaximum value

Probability and StatisticsNumerical MethodsMeasures of Variability / Dispersion2. Variance

Measures how the samples are dispersed.

Probability and StatisticsNumerical MethodsMeasures of Variability / Dispersion3. Standard deviation, s

The positive square root of the variance

Coefficient of variation, Cv

If Cv < 10 – the data are considered clustered, else the data are dispersed

Probability and StatisticsNumerical MethodsMeasures of Shape1. Skewness

A measure of the symmetry of the distribution of the sample

If Sk < 0 – the distribution is skewed to the left (i.e., left tail is longer than right tail)



If Sk = 0 – the distribution is symmetric with respect to the mean, i.e., right and left tails are of equal length (the distribution is called normal or Gaussian)



If Sk > 0 – the distribution is skewed to the right (i.e., right tail is longer than left tail)

Probability and StatisticsNumerical MethodsMeasures of Shape2. Kurtosis

A measure of the height of the distribution

If kurtosis < 0 – the distribution has short height or is almost flat



If kurtosis = 0 – the distribution has the right height



If kurtosis > 0 – the distribution has a high peak

Probability and StatisticsNumerical MethodsMeasures of Data Location1. Quartiles: Q1, Q2, Q3

It is the 25%, 50% and 75% respectively of the data

2. Deciles: D1, D2, D3, … D9

It is the 10%, 20%, 30%,…90% respectively of the data

3. Percentile: P1, P2, P3, … P99

It is the 1%, 2%, 3%,…99% respectively of the data

Probability and StatisticsQuizThe diameter of 36 rivet heads in 1/100 of an inch is given below:

6.72 6.77 6.82 6.70 6.78 6.70 6.62 6.75 6.666.66 6.64 6.76 6.73 6.80 6.72 6.76 6.76 6.686.66 6.62 6.72 6.76 6.70 6.78 6.76 6.67 6.706.72 6.74 6.81 6.79 6.78 6.66 6.76 6.76 6.72

1. Construct a Cumulative Frequency Table2. Determine the Mean, Median and Mode3. Determine the Variance, Standard deviation and the

coefficient of variation

4. Determine the skewness and kurtosis of the distribution and make a conclusion about the shape of the distribution

class interval class boundaries class mark, x i frequency, f i relative freq’y. %

Documents

class boundaries frequency

relative freqy

cumulative frequency