1 3 Measuring Center Spread The Five Number

1. 3 Measuring Center & Spread, The Five Number Summary & Boxplots Describing Quantitative Data with Numbers

1. 3 I can… n n n Calculate and interpret measures of center (mean, median) in context. Calculate and interpret measures of spread (IQR, range, standard deviation) in context. Identify outliers using the 1. 5 IQR rule. Make a boxplot. Selecta ppropriate measures of center and spread. Use appropriate graphs and numerical summaries to compare distributions of quantitative variables.

Measuring Spread n Can two dotplots with the same center and same shape, have different spreads? Explain.

5 -number Summary & Boxplot n n n Minimum Q 1 – Lower Quartile Median Q 3 – Upper Quartile Maximum

Calculations n Measures of Center – Mean – Median n Measures of Spread – Range – IQR – Standard deviation & variance (later) n Outliers

Measuring Center Mean…average value…balance point n Median…typical value…midpoint n n When are the mean and median close to the same?

Resistance n Which is more resistant to extreme values, the mean or median?

Center isn’t enough n The correct mean concentration of hair dye isn’t good enough if some boxes are extremely weak and others are extremely strong. n The correct mean weight of a football isn’t good enough if some are extremely light and some are extremely heavy.

Skewed Distributions n In a right skewed distribution, where is the mean compared to the median? n In a left skewed distribution, where is the mean compared to the median?

Range and Interquartile Range n Range…Hi-Low n Interquartile n See Range…IQR = Q 3 - Q 1 pg. 56 n Which is a more resistant measure of spread, range or IQR? Explain.

Mean or Median? Range or IQR? The mean is sensitive to a few extreme values while the median is not. n A statistician could have his head in an oven and his feet in ice, and he will say that on average he feels fine. n n The range is sensitive to extreme values while the IQR is not.

Example: Amt of fat in Mc. Donald’s beef sandwiches 9, 12, 19, 23, 24, 26, 28, 29, 39, 40, 42

Boxplots Make a boxplot for the Mc. Donald’s beef sandwiches data on amount of fat. n Assess the center, spread, symmetry, and skewness from the boxplot. n Boxplots are a way to visualize the 5 -number summary. n Boxplots do not show the mode like other graphs. n

Comparing Distributions n Boxplots show less detail than histograms or stemplots, so they are best used for side-by-side comparison of more than one distribution.

Outliers 1. 5 x IQR n If an observation falls more than 1. 5 x IQR above third quartile or below the first quartile, it can be considered an outlier. n Outliers may reveal interesting information or they may reveal errors. n

Comparing Quiz Grades n Class A 10, 5, 6, 7, 8, 5, 6, 2 n Class B 2, 10, 4, 2, 5, 1, 10, 9, 7

Comparing Quiz Grades n Min n Q 1 n Median n Q 3 n Max n Outliers? Class A 2 5 6 7 10 Class B 1 2 6 10 10

Comparing Quiz Grades Side-by-Side Boxplots Make the boxplots. Which class did better?

One more measure… Exploring Data- Shape, Outliers, Center, Spread n Graphically Histogram, pie chart, dotplot, stemplot, back to back stemplot, boxplot n Numerically Measures of center mean, mode, median Measures of Spread or Variability range, IQR standard deviation & variance

Standard Deviation Standard deviation measures spread by looking at how far the observations are from the mean. n Be able to interpret the standard deviation…It is roughly the average distance each data value is from the mean of the distribution. n

Standard Deviation Formula The standard deviation is the square root of the average squared difference from the mean.

Comparing Quiz Grades n Class A 10, 5, 6, 7, 8, 5, 6, 2 n Class B 2, 10, 4, 2, 5, 1, 10, 9, 7

Standard deviation with a calculator n Understanding what the standard deviation means is more important than being able to calculate it by hand. n Use s if data is from a sample, and use ∂if data consists of an entire population.

Questions about standard deviation n Standard deviation is a measure of spread about the mean as center, so… -When is the standard deviation 0? -What makes the standard deviation larger?

What to use and when… Mean with standard deviation (not resistant) Median with range & IQR (resistant) Which is best used to describe • symmetric distributions without outliers (such as the Normal distribution)? • skewed distributions? *Numerical summaries do not fully describe the shape of a distribution. Always plot your data!

Variance n Variance is the standard deviation squared. n Standard deviation is the square root of the variance.

Find & interpret the variance for Class A and for Class B n Class A 10, 5, 6, 7, 8, 5, 6, 2 n Class B 2, 10, 4, 2, 5, 1, 10, 9, 7

Exit Slip n The data below indicates the level of phosphate (in milligrams per deciliter) in a patient’s blood over six trips to the hospital. 5. 6 5. 2 4. 6 4. 9 5. 7 6. 4 n Calculate the mean, standard deviation, and variance of the sample size.