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Statistics 100 Lecture Set 6

Statistics 100 Lecture Set 6. Re-cap. Last day, looked at a variety of plots For categorical variables, most useful plots were bar charts and pie charts Looked at time plots for quantitative variables Key thing is to be able to quickly make a point using graphical techniques. Re-cap.

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Statistics 100 Lecture Set 6

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  1. Statistics 100Lecture Set 6

  2. Re-cap • Last day, looked at a variety of plots • For categorical variables, most useful plots were bar charts and pie charts • Looked at time plots for quantitative variables • Key thing is to be able to quickly make a point using graphical techniques

  3. Re-cap • Recall: • A distribution of a variable tells us what values it takes on and how often it takes these values.

  4. Histograms • Similar to a bar chart, would like to display main features of an empirical distribution (or data set) • Histogram • Essentially a bar chart of values of data • Usually grouped to reduce “jitteriness” of picture • Groups are sometimes called “bins”

  5. Histogram • Uses rectangles to show number (or percentage) of values in intervals • Y-axis usually displays counts or percentages • X-Axis usually shows intervals • Rectangles are all the same width

  6. Example (discrete data) • In a study of productivity, a large number of authors were classified according to the number of articles they published during a particular period of time.

  7. Example (discrete data)

  8. Example (continuous data) • Experiment was conducted to investigate the muzzle velocity of a anti-personnel weapon (King, 1992) • Sample of size 16 was taken and the muzzle velocity (MPH) recorded

  9. Constructing a Histogram – continuous data • Find minimum and maximum values of the data • Divide range of data into non-overlapping intervals of equal length • Count number of observations in each interval • Height of rectangle is number (or percentage) of observations falling in the interval • How many categories?

  10. Example • Experiment was conducted to investigate the muzzle velocity of a anti-personnel weapon (King, 1992) • Sample of size 16 was taken and the muzzle velocity (MPH) recorded

  11. What are the minimum and maximum values? • How do we divide up the range of data? • What happens if have too many intervals? • Too Few intervals? • Suppose have intervals from 240-250 and 250-260. In which interval is the data point 250 included?

  12. Histogram of Muzzle Velocity

  13. Interpreting histograms • Gives an idea of: • Location of centre of the distribution • How spread are the data • Shape of the distribution • Symmetric • Skewed left • Skewed right • Unimodal • Bimodal • Multimodal • Outliers • Striking deviations from the overall pattern

  14. Example – mid-term 1 grades (2011) • Was out of 34 + a bonus question (n=344)

  15. Example – mid-term 1 grades (2011) • Too many bins?

  16. Example – mid-term 1 grades (2011) • Too few bins

  17. Example – mid-term 1 grades (2011) • Potential outlier?

  18. Example

  19. Example

  20. Numerical Summaries (Chapter 12) • Graphic procedures visually describe data • Numerical summaries can quickly capture main features

  21. Measures of Center • Have sample of size n from some population, • An important feature of a sample is its central value • Most common measures of center - Mean & Median

  22. Sample Mean • The sample mean is the average of a set of measurements • The sample mean:

  23. Sample Median • Have a set of n measurements, • Median (M) is point in the data that divides the data in half • Viewed as the mid-point of the data • To compute the median: • For sample size “n”, compute position = (n+1)/2 • If position is a whole number, then M is the value at this position of the sorted data • If position falls between two numbers, then M is the value halfway between those two positions in the sorted data

  24. Example • Finding the Median, M, when n is odd • Example: Data = 7, 19, 4, 23, 18

  25. Muzzle Velocity Example • Data (n=16)

  26. Muzzle Velocity Example • Mean:

  27. Muzzle Velocity Example • Median:

  28. Sample Mean vs. Sample Median • Sometimes sample median is better measure of center • Sample median less sensitive to unusually large or small values called • For symmetric distributions the relative location of the sample mean and median is • For skewed distributions the relative locations are

  29. Other Measures of interest • Maximum • Minimum

  30. Percentiles • A percentile of a distribution is a value that cuts off the stated part of the distribution at or below that value, with the rest at or above that value. • 5th percentile: 5% of distribution is at or below this value and 95% is at or above this value. • 25th percentile: 25% at or below, 75% at or above • 50th percentile: 50% at or below, 50% at or above • 75th percentile: 75% at or below, 25% at or above • 90th percentile: 90% at or below, 10% at or above • 99th percentile: ___% at or below, ___% at or above

  31. Percentiles • Can be applied to a population or to a sample • Usually don’t know population • Use sample percentiles to estimate pop. percentiles • Standardized tests often measured in percentiles • Birth statistics often measured in percentiles • First daughter • 10th percentile weight • 25th percentile length • 95th percentile head circumference

  32. Important Percentiles • First Quartile • Second Quartile • Third Quartile

  33. Computing the quartiles • You know how to compute the median • Q1 = • Q3 =

  34. Example • Finding the other quartiles • For Q1, find the median of all values belowM. • For Q3, find the median of all values aboveM. • Example: 4, 7, 18, 19, 23, M=18 • Q1: • Q3: • Example: 4, 7, 12,18, 19, 23, M=15 • Q1: • Q3:

  35. 5 number summary often reported: • Min, Q1, Q2 (Median), Q3, and Max • Summarizes both center and spread • What proportion of data lie between Q1and Q3?

  36. Box-Plot • Displays 5-number summary graphically • Box drawn spanning quartiles • Line drawn in box for median • Lines extend from box to max. and min values. • Some programs draw whiskers only to 1.5*IQR above and below the quartiles

  37. Can compare distributions using side-by-side box-plots • What can you see from the plot?

  38. Example - Moisture Uptake • There is a need to understand degradation of 3013 containers during long term storage • Moisture uptake is considered a key factor in degradation due to corrosion • Calcination removes moisture • Calcination temperature requirements were written with very pure materials in mind, but the situation has evolved to include less pure materials, e.g. high in salts (Cl salts of particular concern) • Calcination temperature may need to be reduced to accommodate salts. • An experiment is to be conducted to see how the calcination temperature impacts the mean moisture uptake

  39. Working Example - Moisture Uptake • Experiment Procedure: • Two calcination temperatures…wish to compare the mean uptake for each temperature • Have 10 measurements per temperature treatment • The temperature treatments are randomly assigned to canisters • Response: Rate of change in moisture uptake in a 48 hour period (maximum time to complete packaging)

  40. Box-Plot

  41. Other Common Measure of Spread: Sample Variance • Sample variance of n observations: • Units are in squared units of data

  42. Sample Standard Deviation • Sample standard deviation of n observations: • Has same units as data

  43. Exercise • Compute the sample standard deviation and variance for the Muzzle Velocity Example

  44. Comments • Variance and standard deviation are most useful when measure of center is • As observations become more spread out, s : increases or decreases? • Both measures sensitive to outliers • 5 number summary is better than the mean and standard deviation for describing (i) skewed distributions; (ii) distributions with outliers

  45. Comments • Standard deviation is zero when • Measures spread relative to

  46. More interpretation of s • Empirical Rule • If a distribution is bell-shaped and roughly symmetric, then • About 2/3 of data will lie within ±1s of • About 95% of data will lie within ±2s of • Usually all data will lie within ±3s of • So you can reconstruct a rough picture of the histogram from just two numbers

  47. Example • Mid-term:

  48. Example • Mid-term: • Empirical rule tells us

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