Chapter 5 Describing Distributions Numerically

Describing a Quantitative Variable using Percentiles Percentile –A given percent of the observations are less than this value. –Ex. 10 th percentile - 10% of the observations of the variable are less than the 10 th percentile. –Ex. 90 th percentile - 90% of the observations of the variable are less than the 90 th percentile.

Important Percentiles Minimum – 0 th percentile Q1 – 25 th percentile (called the first quartile) Median – 50 th percentile Q3 – 75 th percentile (called the third quartile) Maximum – 100 th percentile

Median 50 th percentile –50% of the observations are below the median –50% of the observations are above the median Median is the ______________________ Measures the __________ of the observations

Properties of the Median Which observations affect the median? 73 is an outlier –Does this observation affect the median?

Range Measures spread (variability) Minimum – 0 th percentile Maximum – 100 th percentile Range = _______________________

Properties of the Range Which observations affect the range? 73 is an outlier –Does this observation affect the range?

IQR (Interquartile Range) Measures spread (variability) IQR = Q3 - Q1 Spread of the center 50% of the observations

Finding Q1 and Q3 In general, –Q1 is the _________ of the lower half of the ordered observations. –Q3 is the _________ of the upper half of the ordered observations. Actual calculations from textbook and R may be slightly different.

IQR of Home Runs Per Season for Barry Bonds Order the home runs from smallest to largest Lower Half – –Q1 = 25 Upper Half – –Q3 = 45 IQR = 45 – 25 = 20

Five Number Summary –Min = ____ –Q1 = ____ –Median = _____ –Q3 = _____ –Max = _____

Graph of Five Number Summary Boxplot –Box ___________________________. –Line in the box marks the ____________. –Lines extend out from box to the most extreme data point which is no more than 1.5 times the IQR from the box.

Mean Ordinary average –Add up all observations. –Divide by the number of observations.

Mean Formula –n observations –y 1, y 2, y 3, …, y n are the observations.

Properties of the Mean What effect do the observations have on the mean? 73 is an outlier. What effect does this observation have on the mean?

Standard Deviation Measures spread (variability) “Average” spread from mean. Denoted by letter s.

Standard Deviation

Usually calculate using computer or calculator. –Choose n-1 option on calculator. Do once by hand –Make a table.

Properties of s s ≥ 0 –s = 0 only when all observations are equal. –s > 0 in all other cases. s has the same units as the data.

Properties of s What effect do the observations have on the value of s? 73 is an outlier. What effect does this observation have on the value of s?

Comparison of the Mean and Median Median Mean

Mean vs. Median Mean and Median are generally similar when –Distribution is ________________ Mean and median are generally different when either –Distribution is ________________ –___________ are present.

Influence of Outliers on the Mean and Median Small Example: Income in a small town of 6 people $25,000 $27,000 $29,000 $35,000 $37,000 $38,000 Mean income is $31,830 Median income is $32,000

Influence of Outliers on the Mean and Median –Bill Gates moves to town. $25,000 $27,000 $29,000 $35,000 $37,000 $38,000 $100,000,000 –The mean income is $14,313,000 –The median income is $35,000

Influence of Skewness on the Mean and Median The observations in the tail influence the mean. These observations do not influence the median. –Skewed to the right (large values) ____________________ –Skewed to the left (small values) ____________________

Final Word - Mean vs. Median Always question when means are reported for skewed data –Income –Housing prices –Course grades

Which summaries are the best? Five Number Summary –______________________ Mean and Standard Deviation –______________________ ALWAYS GET A PICTURE OF YOUR DATA.