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Formulae for calculating central tendencies

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• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics Predictions are a type of inferential statistics.
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics Random sampling involves choosing individuals completely at random from a population- for instance putting each student’s name in a hat and drawing one at random. Systematic Sampling involve selecting individuals at regular intervals. For instance, choose every 4 th name on the roll sheet for your class. Stratified sampling makes sure you’re equally representing certain subgroups: for instance, 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, ask a few of your friends.
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics Histograms, bar charts, and pie charts are generally used for categorical data. Frequency polygons are often used for numerical data
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics 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
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics 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.
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics This is the sample variance (the square of the sample standard deviation). Note that we don’t need this formula- we just found S, and the variance is S^2, so we can find this directly.
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics 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).
• Statistics Principles of Engineering TM Unit 4 – Lesson 4.1 - Statistics 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. For some reason the graph does not show the likelihood of zero empty seats, but it is probably quite high, since the bars we see only add to a total of about .50 (50 percent).
• ### Statistics

1. 1. Statistics The collection, evaluation, and interpretation of data
2. 2. Statistics Statistics Descriptive Statistics Describe collected data Inferential Statistics Generalize and evaluate a population based on sample data
3. 3. Data Values that possess names or labels Color of M&Ms, breed of dog, etc. Categorical or Qualitative Data Values that represent a measurable quantity Population, number of M&Ms, number of defective parts, etc. Numerical or Quantitative Data
4. 4. Data Collection Sampling Random Systematic Stratified Cluster Convenience
5. 5. Graphic Data Representation Histogram Frequency Polygons Bar Chart Pie Chart Frequency distribution graph Frequency distribution graph Categorical data graph Categorical data graph %
6. 6. Measures of Central Tendency Most frequently used measure of central tendency Strongly influenced by outliers- very large or very small values Mean Arithmetic average Sum of all data values divided by the number of data values within the array
7. 7. Measures of Central Tendency 48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55 Determine the mean value of
8. 8. Measures of Central Tendency 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)
9. 9. Measures of Central Tendency Determine the median value of Organize the data array from lowest to highest value. 59, 60, 62, 63, 63 48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55 Select the data value that splits the data set evenly. 2, 5, 48, 49, 55, 58, Median = 58 What if the data array had an even number of values? 60, 62, 63, 63 5, 48, 49, 55, 58, 59,
10. 10. Measures of central tendency <ul><li>Usually the highest point of curve </li></ul>Mode Most frequently occurring response within a data array May not be typical May not exist at all Mode, bimodal, and multimodal
11. 11. Measures of Central Tendency 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
12. 12. Data Variation Range Standard Deviation Variance Measure of data scatter Difference between the lowest and highest data value Square root of the variance Average of squared differences between each data value and the mean
13. 13. Range Calculate by subtracting the lowest value from the highest value. 2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63 Calculate the range for the data array.
14. 14. Standard Deviation <ul><li>Calculate the mean . </li></ul><ul><li>Subtract the mean from each value. </li></ul><ul><li>Square each difference. </li></ul><ul><li>Sum all squared differences. </li></ul><ul><li>Divide the summation by the number of values in the array minus 1. </li></ul><ul><li>Calculate the square root of the product. </li></ul>
15. 15. Standard Deviation 2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63 Calculate the standard deviation for the data array. 1. 2. 2 - 47.64 = -45.64 5 - 47.64 = -42.64 48 - 47.64 = 0.36 49 - 47.64 = 1.36 55 - 47.64 = 7.36 58 - 47.64 = 10.36 59 - 47.64 = 11.36 60 - 47.64 = 12.36 62 - 47.64 = 14.36 63 - 47.64 = 15.36 63 - 47.64 = 15.36
16. 16. Standard Deviation 2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63 Calculate the standard deviation for the data array. 3. -45.64 2 = 2083.01 -42.64 2 = 1818.17 0.36 2 = 0.13 1.36 2 = 1.85 7.36 2 = 54.17 10.36 2 = 107.33 11.36 2 = 129.05 12.36 2 = 152.77 14.36 2 = 206.21 15.36 2 = 235.93 15.36 2 = 235.93
17. 17. Standard Deviation 2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63 Calculate the standard deviation for the data array. 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 5. 11-1 = 10 6. 7. S = 22.42
18. 18. Variance <ul><li>Calculate the mean. </li></ul><ul><li>Subtract the mean from each value. </li></ul><ul><li>Square each difference. </li></ul><ul><li>Sum all squared differences. </li></ul><ul><li>Divide the summation by the number of values in the array minus 1. </li></ul>Average of the square of the deviations
19. 19. Variance 2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63 Calculate the variance for the data array.
20. 20. Graphing Frequency Distribution 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. HHH HHT HTH THH HTT THT TTH TTT 3 2 2 2 1 1 1 0 X =1 when? HTT,THT,TTH Toss Outcome X Value
21. 21. Graphing Frequency Distribution The calculated likelihood that an outcome variable will occur within an experiment HHH HHT HTH THH HTT THT TTH TTT 3 2 2 2 1 1 1 0 0 1 2 3 Toss Outcome X value x P(x)
22. 22. Graphing Frequency Distribution 0 1 2 3 x Histogram x P(x)
23. 23. Histogram Open airplane passenger seats one week before departure What information does the histogram provide the airline carriers? What information does the histogram provide prospective customers?