CHAPTER 5 Basic Statistics Data Summary and Presentation: Types of Data Graphical Data Presentation Numerical Data Presentation

Statistic in Engineering Statistics is science of data; the art of conducting studies or experiments through collecting, organizing, summarizing and analyzing data before some conclusion are drawn.

DESCRIPTIVE STATISTICS INFERENTIAL STATISTICS Present quantitative description in manageable form. Graphical representatives and numerical analysis. To make inferences from sample data to reflect the population. Also called inductive reasoning or inductive statistics. Using a sample to draw conclusions about a population

Basic Terms in Statistics Population Entire collection of individuals which are characteristic being studied. Sample A portion, or part of the population interest (a group of subjects selected from a population). Variable Characteristics of the individuals within the population. Observation Value of variable for an element. Data Set - A collection of observation on one or more variables. Name Score Mohd Amirul bin Hamdi 90 Variable Element Observation/ Measurement

Standard error of proportion Characteristic Population Sample Statistical measure Parameter Statistic Size N n Mean Variance Standard deviation Proportion Standard error of mean Standard error of proportion

Data Summary & Presentation

Types of Data: Qualitative & Quantitative Data Deals with numbers. Data which can be measured. Describe by quantity. There are two types of quantitative data:- i. Discrete Data Items or values are countable, can assume only certain values with no intermediate values. The possible values may be fixed (finite) to infinity. ii. Continuous Data Measurements and possible values cannot be counted and can be described using interval on real number line.

Types of Data: Qualitative & Quantitative Data 2) Qualitative Data Deals with descriptions. Can be observed but not measured. Cannot be assume in a numerical value but can be classified into two or more nonnumeric categories. Describe by quality.

TYPES OF DATA Quantitative Qualitative Discrete (e.g, number of houses, cars accidents Continuous (e.g., length, age, height, weight, time) e.g., gender, marital status Qualitative

(can be in quantitative) Measurement Levels and the Appropriate Averages ALL DATA Qualitative data Quantitative data Nominal Car makes, Days of Week, Gender Ordinal (can be in quantitative) TV channel Ranks and title Calendar dates Interval and Ratio Sales ($) Accounts Receivable Market share Mode Median Mean

Grouped Data Vs Ungrouped Data Grouped data - Data that has been organized into groups (into a frequency distribution). Ungrouped data - Data that has not been organized into groups. Also called as raw data.

Graphical Data Presentation

Graphical Data Presentation Data can be summarized or presented in two ways: 1. Tabular 2. Charts/graphs. The presentations usually depends on the type (nature) of data whether the data is in qualitative (such as gender and ethnic group) or quantitative (such as income and CGPA).

Data Presentation of Qualitative Data Tabular presentation for qualitative data is usually in the form of frequency table that is a table represents the number of times the observation occurs in the data. *Qualitative :- characteristic being studied is nonnumeric. Examples:- gender, religious affiliation or eye color. The most popular charts for qualitative data are: 1. bar chart/column chart; 2. pie chart; and 3. line chart.

Types of Graph Qualitative Data

Example 5.1.1: Frequency Table Bar Chart: used to display the frequency distribution in the graphical form. Example 5.1.2: Observation Frequency Malay 33 Chinese 9 Indian 6 Others 2

Pie Chart: used to display the frequency distribution Pie Chart: used to display the frequency distribution. It displays the ratio of the observations Example 5.1.3 : Line chart: used to display the trend of observations. It is a very popular display for the data which represent time. Example 5.1.4 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 10 7 5 39 260 316 142 11 4 9

Data Presentation Of Quantitative Data Tabular presentation for quantitative data is usually in the form of frequency distribution that is a table represent the frequency of the observation that fall inside some specific classes (intervals). *Quantitative : variable studied are numerically. Examples:- balanced in accounts, ages of students, the life of an automobiles batteries such as 42 months). Frequency distribution: A grouping of data into mutually exclusive classes showing the number of observations in each class.

There are few graphs available for the graphical presentation of the quantitative data. The most popular graphs are: 1. Frequency distribution and histogram; 2. frequency polygon 3. Ogive. 4. Stem-and-Leaf plot 5. Box plot.

Weight (Rounded decimal point) Example 5.1.5: Frequency Distribution Histogram: Looks like the bar chart except that the horizontal axis represent the data which is quantitative in nature. There is no gap between the bars. Example 5.1.6: Weight (Rounded decimal point) Frequency 60-62 5 63-65 18 66-68 42 69-71 27 72-74 8

HOW TO CONSTRUCT HISTOGRAM Prepare the frequency distribution table by: Find the minimum and maximum value. Decide the number of classes to be included in your frequency distribution table. Usually 5-20 classes. Too small – may not able to see any pattern. OR Sturge’s rule: Number of classes = . Determine class width, i=(max-min)/num. of class. Determine class limit. Use smallest value for the lower limit of first class. Find class boundaries and class mid points. Count frequency for each class. Draw histogram.

Frequency Polygon: looks like the line chart except that the horizontal axis represent the class mark of the data which is quantitative in nature. Example 5.1.7 : Ogive: line graph with the horizontal axis represent the upper limit of the class interval while the vertical axis represent the cummulative frequencies. Example 5.1.8 :

Stem-and-Leaf Plot : visual way to display a set of data usually large data set Example 5.1.9 : Box Plot Example 5.1.10 : STEM LEAF 2 3 3 3 4 4 5 8 9 9 1 0 0 1 2 3 3 4 5 6 6 8 9 9 2 1 3 Distribution data: skewed to the right with a short tail

Numerical Data Presentation Measures of Central Tendency Measures of Dispersion Measures of Position

Data Summary Summary statistics are used to summarize a set of observations. Two basic summary statistics are measures of central tendency and measures of dispersion. Measures of Central Tendency Mean Median Mode Measures of Dispersion Range Variance Standard deviation Measures of Position Z scores Percentiles Quartiles Outliers Measures of average are also called measures of central tendency and include the mean, median, mode, and midrange. How the data values are dispersed, do the data values cluster around the mean, or are they spread more evenly throughout the distribution? The measures that determine the spread of the data values are called measures of variation, or measures of dispersion. Tell where a specific data value falls within the data set or its relative position in comparison with other data values.

Measures of Central Tendency Mean Mean of a sample is the sum of the sample data divided by the total number sample. Mean for ungrouped data is given by: Mean for group data is given by:

Example 5.1.11 (Ungrouped data): Mean for the sets of data 3,5,2,6,5,9,5,2,8,6 Solution :

Example 5.1.12 (Grouped Data): Use the frequency distribution of weights 100 male students in XYZ university, to find the mean. Weight Frequency 60-62 63-65 66-68 69-71 72-74 5 18 42 27 8

Weight (Class Interval Solution : Weight (Class Interval Frequency, f Class Mark, x fx 60-62 63-65 66-68 69-71 72-74 5 18 42 27 8

Median of ungrouped data: The median depends on the number of observations in the data, n . If n is odd, then the median is the (n+1)/2 th observation of the ordered observations. But if is even, then the median is the arithmetic mean of the n/2 th observation and the (n+1)/2 th observation. Median of grouped data:

Averages (The Median) Single middle value Ordered data The median is the middle value of a set of data once the data has been ordered. Example 5.1.13. Ali hit 11 balls in a golf tournament. The recorded distances of his drives, measured in yards, are given below. Find the median distance for his drives. 85, 125, 130, 65, 100, 70, 75, 50, 140, 95, 70 50, 65, 70, 70, 75, 85, 95, 100, 125, 130, 140 Ordered data Single middle value Median drives = 85 yards

Two middle values so take the mean. Averages (The Median) The median is the middle value of a set of data once the data has been ordered. Example 5.1.14. Ali hit 12 balls at golf tournament. The recorded distances of his drives, measured in yards, are given below. Find the median distance for his drives. 85, 125, 130, 65, 100, 70, 75, 50, 140, 135, 95, 70 50, 65, 70, 70, 75, 85, 95, 100, 125, 130, 135, 140 Ordered data Two middle values so take the mean. Median drive = 90 yards

Weight (Class Interval Cumulative Frequency, F Example 5.1.15 (Grouped Data): The sample median for frequency distribution as in example 5.1.12 Solution: Weight (Class Interval Frequency, f Class Mark, x fx Cumulative Frequency, F Class Boundary 60-62 63-65 66-68 69-71 72-74 5 18 42 27 8 61 64 67 70 73 305 1152 2814 1890 584

Mode Mode of ungrouped data: The value with the highest frequency in a data set. *It is important to note that there can be more than one mode and if no number occurs more than once in the set, then there is no mode for that set of numbers

Mode for grouped data

Weight (Class Interval Cumulative Frequency, F Example 5.1.16 (Ungrouped data) Find the mode for the sets of data 3, 5, 2, 6, 5, 9, 5, 2, 8, 6 Mode = number occurring most frequently = 5 Example 5.1.17 Find the mode of the sample data below Solution: Weight (Class Interval Frequency, f Class Mark, x fx Cumulative Frequency, F Class Boundary 60-62 63-65 66-68 69-71 72-74 5 18 42 27 8 61 64 67 70 73 305 1152 2814 1890 584 23 65 92 100 59.5-62.5 62.5-65.5 65.5-68.5 68.5-71.5 71.5-74.5 Total 6745 Mode class

Measures of Dispersion Range = Largest value – smallest value Variance: measures the variability (differences) existing in a set of data. The variance for the ungrouped data: The standard deviation for the ungrouped data:

Measures of Dispersion The variance for the grouped data: The standard deviation for the grouped data:

A large variance means that the individual scores (data) of the sample deviate a lot from the mean. A small variance indicates the scores (data) deviate little from the mean.

Example 5.1.18 (Ungrouped data) Find the variance and standard deviation of the sample data : 3, 5, 2, 6, 5, 9, 5, 2, 8, 6

Weight (Class Interval Cumulative Frequency, F Example 5.1.19 (Grouped data) Find the variance and standard deviation of the sample data below: Weight (Class Interval Frequency, f Class Mark, x fx Cumulative Frequency, F Class Boundary 60-62 63-65 66-68 69-71 72-74 5 18 42 27 8 61 64 67 70 73 305 1152 2814 1890 584 23 65 92 100 59.5-62.5 62.5-65.5 65.5-68.5 68.5-71.5 71.5-74.5 Total 6745

Weight (Class Interval Cumulative Frequency, F Example 5.1.20 (Grouped data) Find the variance and standard deviation of the sample data below: Weight (Class Interval Frequency, f Class Mark, x Cumulative Frequency, F Class Boundary fx x2 fx2 60-62 63-65 66-68 69-71 72-74 5 18 42 27 8 61 64 67 70 73 23 65 92 100 59.5-62.5 62.5-65.5 65.5-68.5 68.5-71.5 71.5-74.5 305 1152 2814 1890 584 3721 4096 4489 4900 5329 18605 73728 188538 132300 42632 Total 6745 455803