Statistics Statistics Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Statistics

Statistics The collection, evaluation, and interpretation of data Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics The collection, evaluation, and interpretation of data

Statistics Statistics Inferential Statistics Descriptive Statistics Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Statistics Inferential Statistics Generalize and evaluate a population based on sample data Descriptive Statistics Describe collected data Predictions are a type of inferential statistics.

Data Categorical or Qualitative Data Statistics Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Categorical or Qualitative Data Values that possess names or labels Color of M&Ms, breed of dog, etc. Numerical or Quantitative Data Values that represent a measurable quantity Population, number of M&Ms, number of defective parts, etc.

Data Collection Sampling Random Systematic Stratified Cluster Statistics Data Collection Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Sampling Random Systematic Stratified Cluster Convenience Random sampling involves choosing individuals completely at random from a population. For instance, random sampling is practiced when you put each student’s name in a hat and draw one at random. Systematic sampling involve selecting individuals at regular intervals. For instance, choose every 4th name on the roll sheet for your class. Stratified sampling ensures that you equally represent certain subgroups. For instance, you may randomly choose 2 males and 2 females in your class. Cluster sampling involves picking a few areas and sampling everyone in those areas. For instance, sample everyone in the first row and everyone in the third row, but no one else. A convenience sample follows none of these rules in particular. For instance, if you ask a few of your friends for feedback, you are performing a convenience sample.

Graphic Data Representation Statistics Graphic Data Representation Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Histogram Frequency distribution graph Frequency Polygons Frequency distribution graph Bar Chart Categorical data graph Histograms, bar charts, and pie charts are generally used for categorical data. Frequency polygons are often used for numerical data Pie Chart Categorical data graph %

Measures of Central Tendency Statistics Measures of Central Tendency Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Mean Arithmetic average Sum of all data values divided by the number of data values within the array Most frequently used measure of central tendency Strongly influenced by outliers – very large or very small values

Measures of Central Tendency Statistics Measures of Central Tendency Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Determine the mean value of 48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55

Measures of Central Tendency Statistics Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Median Data value that divides a data array into two equal groups Data values must be ordered from lowest to highest Useful in situations with skewed data and outliers (e.g., wealth management)

Measures of Central Tendency Statistics Measures of Central Tendency Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Determine the median value of 48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55 Organize the data array from lowest to highest value. 2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63 Select the data value that splits the data set evenly. If the data array has an even number of values, we take the average (mean) of the two middlemost values. In the example, this is 58.5. Median = 58 What if the data array had an even number of values? 5, 48, 49, 55, 58, 59, 60, 62, 63, 63

Measures of Central Tendency Statistics Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Mode Most frequently occurring response within a data array Usually the highest point of curve May not be typical May not exist at all Modal, bimodal, and multimodal

Measures of Central Tendency Statistics Measures of Central Tendency Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Determine the mode of 48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55 Mode = 63 Determine the mode of 48, 63, 62, 59, 58, 2, 63, 5, 60, 59, 55 Mode = 63 & 59 Bimodal Determine the mode of 48, 63, 62, 59, 48, 2, 63, 5, 60, 59, 55 Mode = 63, 59, & 48 Multimodal

Data Variation Range Standard Deviation Measure of data scatter Statistics Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Measure of data scatter Range Difference between the lowest and highest data value Standard Deviation Square root of the variance

Statistics Range Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Calculate by subtracting the lowest value from the highest value. Calculate the range for the data array. 2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63

Standard Deviation s for a sample, not population Calculate the mean Statistics Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics s for a sample, not population Calculate the mean Subtract the mean from each value and then square it. Sum all squared differences. Divide the summation by the number of values in the array minus 1. Calculate the square root of the product. Note that this is the formula for the sample standard deviation, which statisticians distinguish from the population standard deviation. In practice, only the sample standard deviation can be measured and therefore is more useful for applications.

Standard Deviation Statistics Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Calculate the standard deviation for the data array. 2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63 1. 2. (2 - 47.64)2 = 2083.01 (5 - 47.64)2 = 1818.17 (48 - 47.64)2 = 0.13 (49 - 47.64)2 = 1.85 (55 - 47.64)2 = 54.17 (58 - 47.64)2 = 107.33 (59 - 47.64)2 = 129.05 (60 - 47.64)2 = 152.77 (62 - 47.64)2 = 206.21 (63 - 47.64)2 = 235.93

Statistics Standard Deviation Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Calculate the standard deviation for the data array. 2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63 4. 2083.01 + 1818.17 + 0.13 + 1.85 + 54.17 + 107.33 + 129.05 + 152.77 + 206.21 + 235.93 + 235.93 = 5,024.55 6. 5. s = 22.42

Graphing Frequency Distribution Statistics Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Numerical assignment of each outcome of a chance experiment A coin is tossed 3 times. Assign the variable X to represent the frequency of heads occurring in each toss. Toss Outcome x Value HHH 3 x =1 when? HHT 2 HTH 2 HTT,THT,TTH THH 2 HTT 1 THT 1 TTH 1 TTT

Graphing Frequency Distribution Statistics Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics The calculated likelihood that an outcome variable will occur within an experiment Toss Outcome X value x Px HHH 3 HHT 2 HTH 2 1 THH 2 HTT 1 2 THT 1 TTH 1 3 TTT

Graphing Frequency Distribution Statistics Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Histogram x Px 1 2 Because this is a frequency histogram, the heights of all the bars must add to 1 (1/8 + 3/8 + 3/8 + 1/8 = 1). x 3

Histogram Available airplane passenger seats one week before departure Statistics Histogram Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Available airplane passenger seats one week before departure What information does the histogram provide the airline carriers? percent of the time What information does the histogram provide prospective customers? Airline carriers and passengers can see how many seats will likely be open on a flight one week prior to departure. For instance (looking at the tallest bar) 12 percent of the time there are 5 empty seats. Note that something is wrong with this relative frequency graph! The bars we see only add to a total of about .50 (50 percent), and they should add to 100%. open seats