ADNAN MENDERES UNIVERSITY Faculty of Engineering Measures of

ADNAN MENDERES UNIVERSITY Faculty of Engineering Measures of central tendency and variability MAT 254 Probability & Statistics Week #3 Olcay ÜZENGİ AKTÜRK, Assoc. Prof. Dr.

Data Types • Primary • Ours • Secondary • Not ours • Qualitative data (words) • Blue, short • Quantitative data (numbers) • 1, 2. 5, 10000 • Discrete (countable) • 1 car, 206 students • Continuous (measurable) • 165 cm, 52. 5 kg 2

Sampling Techniques • Probability Sampling (every member of the population has equal chance) • • • Simple Random Sampling (Lottery) Systematic Sampling (every 4 th sample) Stratified Sampling (from each area) Cluster or Area Sampling (form clusters) Multi-stage Sampling (multi-stage) • Non-probability Sampling (samples are selected based on an inclusion rule) 3

Age Frequency 12 2 13 13 14 27 15 4 Presentation of Data Textual Method Tabular Method • Rearrangement from lowest to highest • Stem-and-leaf plot • Frequency distribution table (FDT) • Relative FDT • Cumulative FDT • Contingency Table Stem Leaves 1 7, 8 2 0, 3, 3, 4, 5, 6, 7, 8, 9 3 4, 4, 5, 5, 7, 8, 8, 9, 9, 9 4 2, 3, 4, 4, 5, 6, 6, 8, 9 5 0, 0, 0 Graphical Method • Bar Chart • Histogram • Frequency Polygon • Pie Chart • Less than, greater than Ogive 4

Other Graphical Methods Box Plot (Box and Whisker) Lowest value Highest value Median Upper Quartile Lower Quartile Inter-Quartile Range Scatter Plot 5

How can you represent a huge amount of data (numbers) by using only one (or two) number(s)? q. Minimum of them? q. Maximum of them? q. Average of them? q? ? “CENTRAL TENDENCY” 6

• In statistics, a measure of CENTRAL TENDENCY is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. • The most commonly used statistics for measuring the center of a set of data, arranged in order of magnitude, are the mean, median, and mode. q Mean (average) q Median (middle) q Mode (most) 7

Sample Mean ü The mean (arithmetic mean or average) of a set of data is found by adding up all the items and then dividing by the sum of the number of items. ü The mean of a sample is denoted by (read “x bar”). 2 3 7 7 = 2+3+7+7/4 = 4. 75 8

Sample Mean Staff 1 2 3 4 5 6 7 8 9 10 Salary 1$ 1$ 2$ 2$ 100$ 9

Trimmed Mean A trimmed mean is computed by “trimming away” a certain percent of both the largest and the smallest set of values. For example, the 10% trimmed mean is found by eliminating the largest 10% and smallest 10% and computing the average of the remaining values. 10

Sample Median 2 3 7 7 = 3+7/2 =5 3 7 2 2 3 7 7 3 =3 11

Sample Mode The most repeated value in observations 2 3 7 7 7 unimodal 2 2 3 7 7 2 7 bimodal 12

Sample #1 : 1 51 101 151 201 No mode ! Sample #2 : 99 100 101 102 103 No mode ! Measures of spread or variability ? ? 13

Sample Range The difference between the lowest and the highest value of that sample. 2 3 7 7 The range is 7 -2 = 5 14

Variance & Standard Deviation 15

Variance & Standard Deviation 1 3 5 11 = 1+3+5+11/4 =5 16

Population vs. Sample Commonly used Symbols for a Sample and for a Population. 17

Example from textbook 18

Example from textbook 19

Example from textbook 20

Example from textbook 21

Example from textbook 22

END of LECTURE #3 MAT 254 -02 Probability & Statistics 23