Measures of Position Where does a certain data value fit in relative to the other data values?

N th Place The highest and the lowest 2 nd highest, 3 rd highest, etc. “If I made $60,000, I would be 6 th richest.”

If you know the x valueTo work backward from z to x

Practice computing z scores

Example 3-29: Test Scores A student scored 65 on a calculus test that had a mean of 50 and a standard deviation of 10; she scored 30 on a history test with a mean of 25 and a standard deviation of 5. Compare her relative positions on the two tests. 10 Bluman, Chapter 3 She has a higher relative position in the Calculus class.

Percentiles “What percent of the values are lower than my value?” – 90 th percentile is pretty high – 50 th percentile is right in the middle – 10 th percentile is pretty low If you scored in the 99 th percentile on your SAT, I hope you got a scholarship.

With these salary values again What’s the percentile for a salary of $59,000 ? You can see it’s going to be higher than 50 th.

Example: Finding the percentile

Excel will find the percentile Excel will compute it but slightly differently. PERCENTRANK.EXC(cells, value) For $59,000 Excel gives 0.74 It does some fancy “interpolation” to come up with its results

Quartiles Q 1, Q 2, Q 3 Data values are arranged from low to high. The Quartiles divide the data into four groups. Q2 is just another name for the Median. Q1 = Find the Median of Lowest to Q 2 values Q3 = Find the Median of Q 2 to Highest values It gets tricky, depending on how many values.

Quartiles example 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 Q 2 = median = 50 in the middle. Remove it and split into subsets left and right. Q 1 = median(0, 10, 20, 30, 40) = 20 Q 3 = median(60, 70, 80, 90, 100) = 80

Quartiles example

Quartiles with TI-84 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110 Put values into a TI-84 List Use STAT, CALC, 1-Var Stats

Quartiles in Excel =QUARTILE.INC(cells, 1 or 2 or 3) seems to give the same results as the old QUARTILE function There’s new =QUARTILE.EXC(cells, 1 or 2 or 3) Excel does fancy interpolation stuff and may give different Q1 and Q3 answers compared to the TI-84 and our by-hand methods.

Quintiles and Deciles You might also encounter – Quintiles, dividing data set into 5 groups. – Deciles, dividing data set into 10 groups. Reconcile everything back with percentiles: – Quartiles correspond to percentiles 25, 50, 75 – Deciles correspond to percentiles 10, 20, …, 90 – Quintiles correspond to percentiles 20, 40, 60, 80

Interquartile Range and Outliers

Outliers Example

No-Outliers Example

Outliers: Good or Bad? “I have an outlier in my data set. Should I be concerned?” – Could be bad data. A bad measurement. Somebody not being honest with the pollster. – Could be legitimately remarkable data, genuine true data that’s extraordinarily high or low. “What should I do about it?” – The presence of an outlier is shouting for attention. Evaluate it and make an executive decision.