Lecture PowerPoint Slides Basic Practice of Statistics 7 th Edition

Questions? (logistics, HW, etc.) Questions over readings?

Countries visited

Entry Slip Calculate the mean and median of these numbers: 1, 3, 7, 2, 2

The most common measure of center is the arithmetic average, or mean. Measuring center: the mean To find the mean, (pronounced “x-bar”), of a set of observations, add their values and divide by the number of observations. If the n observations are x 1, x 2, x 3, …, x n, their mean is: or, in more compact notation To find the mean, (pronounced “x-bar”), of a set of observations, add their values and divide by the number of observations. If the n observations are x 1, x 2, x 3, …, x n, their mean is: or, in more compact notation

Measuring center: the median Because the mean cannot resist the influence of extreme observations, it is not a resistant measure of center. Another common measure of center is the median. The median, M, is the midpoint of a distribution, the number such that half of the observations are smaller and the other half are larger. To find the median of a distribution: 1.Arrange all observations from smallest to largest. 2.If the number of observations n is odd, the median M is the center observation in the ordered list. If the number of observations n is even, the median M is the average of the two center observations in the ordered list. 3.You can always locate the median in the ordered list of observations by counting up (n + 1)/2 observations from the start of the list. The median, M, is the midpoint of a distribution, the number such that half of the observations are smaller and the other half are larger. To find the median of a distribution: 1.Arrange all observations from smallest to largest. 2.If the number of observations n is odd, the median M is the center observation in the ordered list. If the number of observations n is even, the median M is the average of the two center observations in the ordered list. 3.You can always locate the median in the ordered list of observations by counting up (n + 1)/2 observations from the start of the list.

Measuring center Key: 1|5 represents a North Carolina worker who reported a 15- minute travel time to work.  Use the data below to calculate the mean and median of the commuting times (in minutes) of 15 randomly selected North Carolina workers.

Comparing the mean and median The mean and median measure center in different ways, and both are useful. The mean and median of a roughly symmetric distribution are close together. If the distribution is exactly symmetric, the mean and median are exactly the same. In a skewed distribution, the mean is usually farther out in the long tail than is the median.

Question: Among the students in this room, is it more likely for the mean credit card debt to be $0 or the median credit card debt to be $0?

Types of quantitative variables Ordinal (shouldn’t use Chapter 2 methods) Scale Ratio (meaningful zero)

Some real datasets

Comments on HW1 Rates are often more useful than raw numbers because they allow for direct comparison. “I got 32 questions right on my Spanish test.” “Last season I scored 35 points.” 50,000 people liked the music video on YouTube. Skew: Where’s the long tail?

Spreadsheet Assignment #1 (copy formulas down!) =SUM(A2, B2) – C2Advanced =SUM(A2:B2) – C2Advanced =A2 + B2 – C2Quick and dirty =SUM(A2 + B2 – C2)Okay, but dangerous =SUM(A2, B2, -C2) =MINUS((A2 + B2), C2) =B:B+C:C-D:D Spreadsheet Assignment #2

Trend Chart (Motion Chart)

Entry Slip: Resistant measures You and nine friends are sitting around calculating statistics on the number of Facebook followers you have. It turns out that you have the most, with 200 followers. You leave, and Ronaldo appears (world record holder for followers), and your friends recalculate the statistics. Which of the following six statistics do not change at all? mean median standard deviation range variation Q3 (third quartile)

Measuring spread: quartiles  A measure of center alone can be misleading.  A useful numerical description of a distribution requires both a measure of center and a measure of spread. To calculate the quartiles:  Arrange the observations in increasing order and locate the median M.  The first quartile, Q 1, is the median of the observations located to the left of the median in the ordered list.  The third quartile, Q 3, is the median of the observations located to the right of the median in the ordered list.

Five-number summary and boxplots Consider a second travel times data set, these from New York. Find the five-number summary and construct a boxplot. M = 22.5 Q 3 = 42.5 Q 1 = 15 Min= Max=85

Figure 2.1, The Basic Practice of Statistics, © 2015 W. H. Freeman

Figure 2.2, The Basic Practice of Statistics, © 2015 W. H. Freeman

Measuring s pread: standard deviation The most common measure of spread looks at how far each observation is from the mean. This measure is called the standard deviation. The variance, s 2, of a set of observations is an average of the squares of the deviations of the observations from their mean. In symbols, the variance of the n observation s x 1, x 2, x 3, …, x n, is Again, more briefly: The standard deviation, s, is the square root of the variance, s 2.

Sample Population =VAR, =VARA, =STDEV, =STDEVA =VARP, =VARPA, =STDEVP, =STDEVPA

Calculating the standard d eviation xixi (x i -mean)(x i -mean) =102(102) 2 = =-58(-58) 2 = =32(32) 2 = =-98(-98) 2 = =22(22) 2 = 484 Sum=? 3) Square each deviation. 4) Find the “average” squared deviation. Calculate the sum of the squared deviations divided by (n-1)…this is called the variance. 5) Calculate the square root of the variance…this is the standard deviation. “Average” squared deviation = 24,880/(5 – 1) = This is the variance. Standard deviation = square root of variance =

Figure 2.3, The Basic Practice of Statistics, © 2015 W. H. Freeman

Which set of numbers has the highest variation and standard deviation? (Note: All of these have the same mean.)set of numbers 15 Constant Edges Edges II Wide vary Close vary Center and edge

Is it better to have low or high s? Factory measurements of a car part Personalities of friends Batting average per game Between group variance – control drug vs. experimental drug Within group variance – different people’s reactions to taking a specific drug or dosage

Is it better to have low or high s? Factory measurements of a car part (low) Personalities of friends (high) Batting average per game (depends?) Between group variance – control drug vs. experimental drug (high) Within group variance – different people’s reactions to taking a specific drug or dosage (low)

Properties of s n – 1 is called the degrees of freedom. s measures variability about the mean and should be used only when the mean is chosen as the measure of center. s is always zero or greater than zero. As the observations become more variable about their mean, s gets larger. Unlike variance, s has the same units of measurement as the original observations. s is not resistant.

Option 1 Center: Mean ( ) Spread: Standard deviation (s) Outliers removed Symmetric data Choosing measures of center and spread Option 2 Center: Median Spread: Q1 and Q3 (Five-number summary) Skewed or outliers

U.N. Millennial Goals

Figure 2.6, The Basic Practice of Statistics, © 2015 W. H. Freeman