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Describing Distributions

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### Section 2

1. 1. Describing Distributions with Numbers
2. 2. VOCABULARY <ul><li>Mean – the average of all observations </li></ul><ul><li>Median – the middle number </li></ul><ul><ul><li>Put numbers in order smallest to largest </li></ul></ul><ul><ul><li>If # of observations is odd, M is the center </li></ul></ul><ul><ul><li>If # of observations is even, M is the mean of the two center observations </li></ul></ul>
3. 3. DIFFERENCES BETWEEN MEAN AND MEDIAN <ul><li>Median is resistant to extreme observations </li></ul><ul><ul><li>3 5 8 10 14 17 100 </li></ul></ul><ul><ul><li>Median = 10 </li></ul></ul><ul><li>Mean is not resistant to extreme observations </li></ul><ul><ul><li>5 8 10 14 17 100 </li></ul></ul><ul><ul><li>Mean = 22.4286 </li></ul></ul>
4. 4. SPREAD OR VARIABILITY <ul><li>Range – shows the full spread of the data </li></ul><ul><ul><li>Largest value minus smallest value (outliers can change this value drastically) </li></ul></ul><ul><li>Quartiles – marks out the quarters of the data (middle of the halves) </li></ul>
5. 5. QUARTILES <ul><li>First Quartile – lies one-quarter of the way up the list of data. </li></ul><ul><li>Third Quartile – lies three-quarters of the way up the list of data. </li></ul>
6. 6. CALCULATING THE QUARTILES <ul><li>Arrange the observations in increasing order </li></ul><ul><li>Locate the median M </li></ul><ul><li>The first quartile Q 1 is the median of the observations lying to the left of the overall median (not counting the median value) </li></ul><ul><li>The third quartile Q 3 is the median of the observations lying to the right of the overall median </li></ul>
7. 7. MORE WITH QUARTILES <ul><li>IQR – the interquartile range is the distance between the first and third quartiles. </li></ul><ul><ul><li>Used to determine if an observation is an outlier </li></ul></ul><ul><ul><li>Observation is called an outlier if it falls more than 1.5 x IQR below Q 1 or above Q 3 . </li></ul></ul>
8. 8. FIVE NUMBER SUMMARY <ul><li>Minimum </li></ul><ul><li>Q 1 (First Quartile) </li></ul><ul><li>Median </li></ul><ul><li>Q 3 (Third Quartile) </li></ul><ul><li>Maximum </li></ul><ul><ul><li>This summary defines the boxplots. </li></ul></ul>
9. 9. BOXPLOTS <ul><li>Regular boxplot does not distinguish outliers </li></ul><ul><li>Modified boxplot distinguishes outliers by plotting them as points (use this one!!!) </li></ul>
10. 10. ASSIGNMENT <ul><li>Page 34-36 #1.24 – 1.30 </li></ul><ul><li>Page 37-42 #1.31 – 1.34 </li></ul><ul><li>1997 AP FR Q#1 </li></ul><ul><li>2001 AP FR Q#1 </li></ul>
11. 11. STANDARD DEVIATION <ul><li>Measures spread by looking at how far the observations are from the mean </li></ul><ul><ul><li>Most common numerical method of describing spread </li></ul></ul>
12. 12. VOCABULARY FOR STANDARD DEVIATION <ul><li>Variance – the average of the squares of the deviations of the observations from their mean </li></ul><ul><li>Standard deviation (s) – the square root of the variance </li></ul><ul><li>(Note: The sum of the errors is zero!!!) </li></ul>
13. 13. EXAMPLE <ul><li>Use the given data to answer the questions: </li></ul><ul><li>1792 1666 1362 1614 1460 1867 1439 </li></ul><ul><li>1. Find the mean </li></ul><ul><li>2. Find the variance </li></ul><ul><li>3. Find the standard deviation </li></ul>
14. 14. EXAMPLE <ul><li>Use the given data to answer the questions: </li></ul><ul><li>1792 1666 1362 1614 1460 1867 1439 </li></ul><ul><li>1. Find the mean </li></ul><ul><li>2. Find the variance </li></ul><ul><li>3. Find the standard deviation </li></ul>
15. 15. PROPERTIES OF STANDARD DEVIATION <ul><li>s measures spread about the mean and should be used only when the mean is chosen as the measure of center </li></ul><ul><li>s = 0 only when there is no spread. This happens only when all observations have the same value </li></ul><ul><li>s, like the mean, is strongly influenced by extreme observations (non-resistant) </li></ul>
16. 16. ASSIGNMENT <ul><li>Page 43-47 #1.35 – 1.37 </li></ul><ul><li>Page 48 #1.38, 1.41, 1.43, 1.46 </li></ul><ul><li>Summary Statistics Worksheet </li></ul><ul><li>Prepare for Quiz 1.2 </li></ul>