Basic Statistics q Statistics in Engineering collect organize

Basic Statistics q Statistics in Engineering (collect, organize, analyze, interpret) q Collecting Engineering Data q Data Presentation and Summary Ø Types of Data Ø Graphical Data Presentation Ø Numerical Data Presentation q Probability Distributions Ø Discrete Probability Distribution Ø Continuous Probability Distribution

Collecting Engineering Data Direct observation Experiments Ø better way to produce data Surveys Ø depends on the response rate Personal Interview: Ø higher expected response rate and Ø fewer incorrect respondents

Population: the entire collection of objects or outcomes about which data are collected. Sample: subset of the population containing the observed objects or the outcomes. Parameter: Summary measure about population, . Statistics: Summary measure about sample, . Population vs Sample Parameter vs Statistics

Statistics can be divided into two. 1) Descriptive statistics: describe basic features of data by providing simple summaries about the sample and measures in a form of suitable graphical or numerical analysis. Graphical representatives: v stem-and-leaf plot v line chart v histogram v boxplot. Numerical analyses: v measure of central tendency v measure of dispersion v measure of position. 2) Inferential statistics: draw a conclusion about sample data that would represent an actual population.

Types of Data Qualitative/ Categorical Data Qualitative vs Quantitative/ Numeric Data i. Deals with descriptions. ii. Data can be observed but not measured. i. Deals with numbers. ii. Data which can be measured. i. ii. iii. iv. v. Defect or no defect Gender Ethnic group Colors Textures Income CGPA Diameter Weight cost The most popular charts for qualitative data : Ø bar chart/column chart Ø pie chart Ø line chart. Ø Ø Ø histogram frequency polygon ogive box plot stem and leaf plot

Discrete vs Continuous q Quantitative variables can be further classified as discrete or continuous. q Discrete variables are usually obtained by counting. There a finite or countable number of choices available with discrete data. You can't have 2. 63 people in the room. q Continuous variables are usually obtained by measuring. Length, weight, and time are all examples of continous variables.

Grouped Vs Ungrouped Data q Ungrouped/raw data - Data that has not been organized into groups. q Grouped data - Data that has been organized into groups (into a frequency distribution). v Frequency distribution: A grouping of data into mutually exclusive classes showing the number of observations in each class. Ungrouped data 1. 0, 1. 1, 1. 2, 1. 0, 1. 1, 1. 3, 1. 2, 1. 1, 1. 0, 1. 2, 1. 3, 1. 4, 1. 2, 1. 1, 1. 0, 1. 2, 1. 3, 1. 4, 1. 0 Group data Class boundaries Frequency 0. 95 – 1. 15 10 1. 15 – 1. 35 9 1. 35 – 1. 55 2

Example: q About 50 Uni. MAP students were asked about their background and the results are as follows. Display your data in suitable form. Respondent Gender Code used: Gender: 1 = male, 2 = female Ethnic group: 1 = Malay, 2 = Chinese, 3 = Indian, 4 = others 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 1 1 1 1 2 2 2 1 Ethnic Family Group Income 1 1 1 4 3 1 2 1 1 1 3 1 1 2 1 3 1000 1600 8000 1360 800 1250 1200 3000 4500 3000 2380 800 2000 1000 3500 1600 1803 3000 1400 3000 4780 4300 2500 CGPA Respondent Gender 3. 00 3. 37 3. 59 2. 50 3. 19 2. 96 3. 65 3. 04 2. 80 3. 39 3. 16 3. 67 3. 40 3. 10 3. 31 3. 80 3. 16 2. 84 3. 35 3. 20 3. 00 2. 80 2. 78 2. 90 3. 02 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 Ethnic Family CGPA Group Income 1 1 1 3 2 1 1 1 1 1 4 1 1 1 3 1 2 1 1 3 1900 1000 1500 2000 8000 2000 1570 7000 1000 3000 1980 1200 2670 2000 3596 2000 5000 1500 2500 1500 2. 82 3. 02 3. 47 3. 60 3. 41 3. 23 3. 25 3. 39 3. 20 3. 01 2. 98 3. 45 3. 13 3. 30 2. 60 2. 89 2. 90 3. 70 3. 11 3. 34 3. 82 3. 61 3. 25 3. 85 3. 67

Frequency table graphical presentation of qualitative data Observation Frequency Malay 33 Chinese 9 Indian 6 Others 2 q. Bar Chart: used to display the frequency distribution in graphical form.

q Pie Chart: used to display the frequency distribution. It displays the ratio of the observations 4% 12% Malay 18% Chinese Indian 66% Others q Line chart: used to display the trend of observations. It is a very popular display for the data which represent time. Jan 10 Feb 7 Mar 5 Apr 10 May Jun 39 7 Jul 260 Aug 316 Sep 142 Oct 11 Nov 4 Dec 9

graphical presentation of quantitative data v 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.

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

Data Summary statistics are used to summarize a set of observations. a) Measures of Central Tendency q Mean q Median q Mode b) Measures of Dispersion q Range q Variance q Standard deviation c) Measures of Position q Z scores q Percentiles q Quartiles q Outliers

a) Measures of Central Tendency 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 2 (Ungrouped data): Mean for the sets of data 3, 5, 2, 6, 5, 9, 5, 2, 8, 6 Solution :

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 n 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:

Example 4 (Ungrouped data): n is odd Find the median for data 4, 6, 3, 1, 2, 5, 7 ( n = 7) Rearrange the data : 1, 2, 3, 4, 5, 6, 7 (median = (7+1)/2=4 th place) Median = 4 n is even Find the median for data 4, 6, 3, 2, 5, 7 (n = 6) Rearrange the data : 2, 3, 4, 5, 6, 7 Median = (4+5)/2 = 4. 5

Mode • Mode of ungrouped data: q The value with the highest frequency in a data set. q 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 Find the mode for the sets of data 3, 5, 2, 6, 5, 9, 5, 2, 8, 6 Mode = number occurring most frequently = 5

b) 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: For sample For population

The variance for the grouped data: For sample or For population or

Standard deviation: the positive square root of the variance is the standard deviation 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 8 (Ungrouped data) Find the variance and standard deviation of the sample data : 3, 5, 2, 6, 5, 9, 5, 2, 8, 6

Exercise 4 (submit on Thursday) The following data give the sample number of i. Pads sold by a mail order company on each of 30 days. (Hint : 5 number of classes) 8 25 11 15 29 22 10 5 22 13 26 16 18 12 9 23 14 19 23 20 16 27 9 17 21 26 20 16 21 14 a) Construct a frequency distribution table. b) Find the mean, variance and standard deviation, mode and median. c) Construct a histogram.

Rules of Data Dispersion By using the mean and standard deviation, we can find the percentage of total observations that fall within the given interval about the mean. Empirical Rule Applicable for a symmetric bell shaped distribution / normal distribution. There are 3 rules: i. 68% of the data will lie within one standard deviation of the mean, ii. 95% of the data will lie within two standard deviation of the mean, iii. 99. 7% of the data will lie within three standard deviation of the mean,

Example 10 The age distribution of a sample of 5000 persons is bell shaped with a mean of 40 yrs and a standard deviation of 12 yrs. Determine the approximate percentage of people who are 16 to 64 yrs old. Solution: Approximately 68% of the measurements will fall between 28 and 52, approximately 95% of the measurements will fall between 16 and 64 and approximately 99. 7% to fall into the interval 4 and 76.

c) Measures of Position To describe the relative position of a certain data value within the entire set of data. q z scores q Percentiles q Quartiles q Outliers

Quartiles q Divide data sets into four equal parts where each part account about 25% of data distribution. Minimum value 25% of data Q 1 Q 2 25% of data Q 3 25% of data Maximum value 25% of data

Find Q 1, Q 2, and Q 3 for the following data 15, 13, 6, 5, 12, 50, 22, 18 Step 1: Arrange the data in order 5, 6, 12, 13, 15, 18, 22, 50 Step 2: Find the median (Q 2) 5, 6, 12, 13, 15, 18, 22, 50 ↑ Q 2=(13+15)/2=14 Step 3 Find the median of the data values less than 14. 5, 6, 12, 13 ↑ Q 1 = (6+12)/2=9 Step 4 Find the median of the data values greater than 14 15, 18, 22, 50 ↑ Q 3=(18+22)/2=20

Example: 5, 8, 4, 6, 3, 8 (n=7) 1. Arrange the data in order form: 3, 4, 4, 5, 6, 8, 8 2. Q 1: Find the median of the data values less than 5. 3, 4, 4 Q 1: Find the median of the data values greater than 5. 6, 8, 8 Therefore,

Exercise: The following data represent the number of inches of rain in Chicago during the month of April for 10 randomly years. 2. 47 3. 94 4. 11 5. 22 1. 14 4. 02 3. 41 1. 85 0. 97 Determine the quartiles.

Exercise: The following data represent the number of inches of rain in Chicago during the month of April for 10 randomly years. 2. 47 3. 94 4. 11 5. 22 1. 14 4. 02 3. 41 1. 85 0. 97 Determine the quartiles. Answer:

Outliers Extreme observations Can occur because of the error in measurement of a variable, during data entry or errors in sampling.

Checking for outliers by using Quartiles Step 1: Determine the first and third quartiles of data. Step 2: Compute the interquartile range (IQR), . Step 3: Determine the fences. Fences serve as cut off points for determining outliers. needed for identifying extreme values in the tails of the distribution: Step 4: If data value is less than the lower fence or greater than the upper fence, considered outlier. A point beyond an outer fence is considered extreme outlier.

Example 2. 47 3. 94 4. 11 5. 22 1. 14 4. 02 3. 41 1. 85 0. 97 Determine whethere are outliers in data set.

Arrange data in ascending form: 0. 97, 1. 14, 1. 85, 2. 47, 3. 41, 3. 94, 3. 97, 4. 02, 4. 11, 5. 22 Follow the steps to find quartiles 0. 97, 1. 14, 1. 85, 2. 47, 3. 41 3. 94, 3. 97, 4. 02, 4. 11, 5. 22 Since all the data are not less than -3. 0675 and not greater than 8. 2725, then there are no outliers in the data

Boxplot (Graphical presentation for quantitative data) The five-number summary can be used to create a simple graph called a boxplot. Minimum Q 1 Median Q 3 Maximum Form the boxplot, you can quickly detect any skewness in the shape of the distribution and see whethere any outliers in the data set. Outlier Lower fence Upper fence

The Five Number Summary Compute the five-number summary and construct the box plot of the data 2. 47 1. 14 3. 97 4. 02 3. 94 3. 41 4. 11 1. 85 5. 22 0. 97

- The distribution is skewed to the left

Interpreting Boxplot - symmetric - Left skewed or negatively skewed: the tail is skewed to the left - Right skewed or positively skewed: the tail is skewed to the right

Mean/Median Versus Skewness Mean < Median < Mode Mean > Median > Mode Mean = Median = Mode

STEM-AND-LEAF 41 Another technique that is used to present quantitative data is the stem-and-leaf plot. An advantage of a stem-and-leaf over a frequency distribution is that by preparing stem-and-leaf, we do not lose information on individual observations. A stem-and-leaf only for quantitative data. In a stem-and-leaf display of quantitative data, each value is divided into two portions; a stem and leaf. The leaves for each stem are shown separately in a display.

42 • Stem-and-leaf plot display a set of data usually large data set. • Stem and leaf plots emphasize place value. Stem is for the largest place value(s) of a number and leaf is the smallest place value of a number in your data set. Step 1: Find the least and the greatest number in the set of data Step 2: Make two columns with titles STEM and LEAF. Step 3: Write the digits that form the stem in the STEM column Step 4: Write the digits that form the leaf for each number in the LEAF column across from the STEM of the number.

Example: : The following are the scores of 30 college students on a statistics test. 75 52 80 96 65 79 71 87 93 95 69 72 81 61 76 86 79 68 50 92 83 84 77 64 71 87 72 92 57 98 Ø For the score of the first student, which is 75, 7 is the stem and 5 is the leaf. For the score of the second student, which 52, 5 is the stem and 2 is the leaf. Ø Observed from data, the stems for all scores are 5, 6, 7, 8 and 9 because all scores lie in the range 50 to 98. Ø After we have listed the stems, we read the leaves for all scores and record them next to the corresponding stems at the right side of the vertical line.

44 Now we read all the scores and write the leaves on the right side of the vertical line in the rows of corresponding stems. By looking at the stem-and-leaf display of test scores, we can observed how the data values are distributed. For example, the stem 7 has the highest frequency, followed by stems 8, 9, 6 and 5. The leaf for each stem of the stem-and-leaf display of test scores are rank in increasing order and presented as below : Stem Leaf 5 0 2 7 6 1 4 5 8 9 7 1 1 2 2 5 6 7 9 9 8 0 1 3 4 6 7 7 9 2 2 3 5 6 8 The distribution of data seems skewed to the left tail * Analyze – There are 9 out of 30 college students score between 71 and 79. Ranked stem-and-leaf display of test scores.

What you MUST know? Define statistics and its application in engineering. Explain the concept of population and sample. Compute and interpret the measures of central tendency (MCT), measures of dispersion (MD) and measures of position (MP). Construct and interpret several graphical presentation (histogram, box plot, stem and leaf plot). Explain how graphical presentation are used to compare two or more sets of data. Compare MCT, MD and MP for two or more sets of data.