Chapter 1 basic statistics
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Chapter 1 Basic Statistics. FARAH ADIBAH ADNAN ENGINEERING MATHEMATICS INSTITUTE (IMK). CHAPTER 1. Basic Statistics Statistics in Engineering Collecting Engineering Data Data Summary and Presentation Probability Distributions - Discrete Probability Distribution

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Chapter 1 Basic Statistics

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Chapter 1Basic Statistics

FARAH ADIBAH ADNAN

ENGINEERING MATHEMATICS INSTITUTE (IMK)


CHAPTER 1

  • Basic Statistics

    • Statistics in Engineering

    • Collecting Engineering Data

    • Data Summary and Presentation

    • Probability Distributions

      - Discrete Probability Distribution

      - Continuous Probability Distribution

    • Sampling Distributions of the Mean and Proportion


Statistics in engineering

  • Statistics - area of science that deals with collection, organization, analysis, and interpretation of data.

  • Statistics - deals with methods and techniquesthat can be used to draw conclusions about the characteristics of a large number of data points, commonly called a populationby using a smaller subset of the entire data called sample.

  • Because many aspects of engineering practice involve working with data, obviously some knowledge of statistics is important to an engineer.


  • Specifically, statistical techniques can be a powerful aid in designing new products and systems, improving existing designs, and improving production process.

  • The methods of statistics allow scientists and engineers to design valid experiments and to draw reliable conclusions from the data they produce


Collecting Engineering Data

  • Direct observation

    The simplest method of obtaining data.

    Advantage: relatively inexpensive.

    Disadvantage: difficult to produce useful information since it does not consider all aspects regarding the issues.

  • Experiments

    More expensive methods but better way to produce data.

    Data produced are called experimental.

  • Surveys

    Most familiar methods of data collection.

    Depends on the response rate.

  • Personal Interview

    Has the advantage of having higher expected response rate.

    Fewer incorrect respondents.


Data Presentation

  • Data can be categorized into two :-

    - Qualitative data - qualitative attributes

    - Quantitative data - quantitative attributes

  • Two sources of data :-

    - Primary ( eg. Questionnaire, Telephone Interview)

    - Secondary (eg. Internet, Annual Report)

    Data should be summarized in more informative way such as graphical, tables or charts.


Data Presentation

Data can be summarized or presented in two ways:

1) Tabular

2) Charts/graphs.

Data Presentation of Qualitative Data

1) Frequency Distribution Table - represents the number of times the observation occurs in the data.

Example: Ethnic Group

Observation Frequency

Malay33

Chinese9

Indian6

Others2


2) Charts for qualitative data are:

Pie Chart : Gender

Bar Chart : Ethnic Group

Line Chart : Number of Sandpipers from Jan 1989 – Dec 1989


Data Presentation of Quantitative Data

1) Frequency Distribution Table – list all classes and the number of values that belong to each class.


  • This formula will be used to form frequency distribution table, from raw data.

    Class - an interval that includes all the values that fall within two numbers; the lower and upper class (class limit).

    Class Boundary - the midpoint of the upper limit of one class and the lower limit of the next class.

    Class Width/Size/Interval ,c -difference between the two boundaries of a class . Formula :

    C = Upper boundary – Lower Boundary

    Class Midpoint/Mark, x – formula:

    (Lower Limit + Upper Limit)/2


How to Form Frequency Distribution Table

  • Decide the number of classes to be used.

  • Determine class width:

  • When the number of classes are given,

    Class width =

  • When the number of classes are not given,

    Class width =

    where the number of class =

  • Don’t forget to always round up to the nearest whole number when

    dealing with class width/interval.

  • Any convenient number that is equal to or less than the smallest values in the data set can be used as the lower limit of the first class.


  • 25 11 15 29 22 10 5 1721

  • 13 26 16 18 12 9 26 2016

  • 23 14 19 23 20 16 27 9 2114

Example:

The following data give the total number of iPods sold by a mail order company on each of 30 days. Construct a frequency distribution table. (Hint: 5 number classes).

Solution:

Number of classes = 5

Class width =


Frequency Distribution Table


Histogram: Student’s CGPA

Polygon : Student’s CGPA

2) Graph for quantitative data are:

Ogive: Student’s CGPA


Data Summary

Summary statisticsare used to summarize a set of observations.

Two basic summary statistics are

1) Measures of central tendency

- Mean

- Median

- Mode

2) Measures of dispersion

- Range

- Variance

- Standard deviation


Measures of Central Tendency

1) Mean ,( )

  • Mean of a sample ( ) or population ( ) is the sum of the sample data divided by the total number sample.

  • Mean for ungroup data is given by:

    Sample:

    Population:

  • Mean for group datais given by:

    Sample: Population:

    where f = class frequency; x = class mark (mid point)


Example:

1) Find the mean for the set of data 4, 6, 3, 1, 2, 5, 7.

Solution:

2) Find the mean of the frequency distribution table below.


(f)

(x)

Solution:

Therefore, the mean of frequency distribution above is:


2) Median, ( )

  • Median is the middle value of a set of observations arranged in ascending order and normally is denoted by ( ).

  • Median for ungrouped data:

    - The median depends on the number of observations in the data, .

    - If is odd, then the median is the th observation of the ordered observations / middle value.

    - If is even, then the median is the average of the 2 middle values ( th observation and the th observation).


  • Median for grouped data / frequency of distribution.

    The median of frequency distribution is defined by:

    where,

    = the lower class boundary of the median class;

    = the size of the median class interval;

    = the sum of frequencies of all classes lower than the median class;

    = the frequency of the median class.


Example:

1) Find the median for the set of data 4, 6, 3, 1, 2, 5, 7, 3.

Solution:

Arrange in order of magnitude : 1,2,3,3,4,5,6,7.

As n = 8 (even), the median is the mean of the 4th and 5th value.

Therefore, the median is 3.5

2) Find the median of the frequency distribution table below.


Cumulative

Frequency

Solution:

To determine median class:

So, the median class falls in class 3.00 – 3.25.


3) Mode, ( )

  • The mode of a set of observations is the observation with the highest frequencyand is usually denoted by ( ). Sometimes mode can also be used to describe the qualitative data.

    *Note:

  • If data set with only 1 value that occur with the highest frequency, therefore it has 1 mode and it is called unimodal data.

  • If data set has 2 measurements with highest frequency, therefore it has 2 modes and known as bimodal data.

  • If data set has more than 2 measurements with highest frequency, so the data set contains more than 2 modes and said to be multimodal data.


  • For ungrouped data:

    - Defined as the value which occurs most frequent.

    Example:

    The mode for data 4,6,3,1,2,5,7,3 is 3.

  • For grouped data

    When data has been grouped into classes and a frequency curve is drawn to fit the data, the mode is the value of corresponding to the maximum point on the curve.


- Determining the mode using formula.

where

= the lower class boundary of the modal class;

= the size of the modal class interval;

= the difference between the modal class frequency and the class before it;and

= the difference between the modal class frequency and the class after it.

Note: The class which has the highest frequency is called the modal class.


Example:

Find mode of the frequency distribution table below.

Solution:


Measures of Dispersion

  • The measure of dispersion/spread is the degree to which a set of data tends to spread around the average value.

  • It shows whether data will set is focused around the mean or scattered.

  • The common measures of dispersion are:

    1) Range

    2) Variance

    3) Standard deviation

  • The standard deviation actually is the square root of the variance.

  • The sample variance is denoted by s2 and the sample standard deviation is denoted by s.


1) Range

  • Simplest measure of dispersion.

  • Apply for both group & ungroup data.

    Ungroup data:

    Formula: Range = Largest value – Smallest value

    Group data:

    Formula: Range = Largest value (class limit) – Smallest value

    (class limit)

    Example:

    Solution:

    Range = Largest Value – Smallest Value

    = 267, 277 – 49, 651 = 217, 626 square miles.


2) Variance, ( )

  • Measures the variability in a set of data.

  • The variance for the ungrouped data:

    Sample: Population:

  • The variance for the grouped data:

    Sample:

    Population:


Example:

The variance for grouped data :

Solution:


2) Standard Deviation, ( )

  • The positive square root of the variance is the standard deviation.

  • A larger value of the standard deviation – the values of the data set are spread relatively large from the mean.

  • A lower value of the standard deviation – the values of the data set are spread relatively small from the mean.

  • The standard deviation for the ungrouped data:

    Sample: Population:


  • The standard deviation for grouped data:

    Sample:

    Population:

    Example:

    From previous example.


exercise

The final results in business statistics of 40 students are recorded as below

  • Present the data in frequency distribution table.

  • Construct a histogram

  • Calculate mean, median, mode, variance and std deviation.


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