Exam 3: Numerical Methods for Describing Data Distributions

For any given data set, the median must be greater than or equal to the lower quartile, and less than or equal to the upper quartile.

Multiple sclerosis (MS) is an autoimmune disease that affects the brain and spinal cord. MS affects women more than men, and is most commonly diagnosed between the ages of 20 and 40. MS sufferers typically have problems of coordination and walking. The standard care for some sufferers is a plastic device affixed to the lower leg and ankle. In a recent test of this device, volunteers with MS were randomly assigned to two experimental groups. The volunteers in one group used the plastic device; the volunteers in the other group did not use the plastic device. Each individual was timed as he or she climbed a set of stairs. Times, measured without the device (ND) and with the device (D), are presented below. Stairs ND 9.3 8.4 8.6 6.9 17.7 12.4 18.6 14.2 7.9 6.9 8.4 9.4 12.7 Stairs D 7.9 8.2 9.0 7.2 15.3 13.5 16.6 15.7 7.7 7.5 8.4 9.5 12.7 -Construct a back-to-back stem-and-leaf display that will allow comparison of the times on the stair-climbing task.

Construct a back-to-back stem-and-leaf display that will allow comparison of the times on the walking task.

By definition, a data point is an outlier if it is mo re than 3.0 iqr away from the closest quartile.

The forces that determine the size of groups ("swarms") of social insects and the rates at which they grow are not well understood. Biologists have observed large variability in the size of swarms across species. In a study of the social wasp, Polybia occidentalis, investigators dismantled a nest of these insects and marked a few insects for future identification in new swarms. Twenty-five days after dismantling the original swarm they had located new swarms of wasps from the original colonies. The data below present the numbers of adults and workers in the new swarms in two different years. "Drones" are fertile males that mate with the queen. "Workers" are infertile females that labor in the nest and defend the colony. First Year Swarm \# Number of Drones Number of Workers Second Year Swarm \# Number of Drones Number of Workers 1 598 597 1 355 339 2 24 21 2 691 673 3 567 557 3 278 252 4 371 325 4 719 669 5 279 260 5 152 129 6 44 35 6 156 140 7 42 41 7 41 35 8 126 107 9 108 101 10 79 64 -Construct comparative box plots of the numbers of drones in the swarms in the first and second years.

Deviations from the mean are always positive.

The variance is the positive square root of the standard deviation.

Engineers initially surveyed the Territory of Iowa in the 1830 's, and they were very careful to take note of the trees and vegetation. The sample of Ash tree diameters from the original survey of what is now Linn County is presented in the stem-and-leaf display below. The display uses five lines for each stem. Thus, "1t|" is the stem for diameters of 12 and 13 , "1f|" for 14 and $15 , " 1 \mathrm {~s} \mid "$ for 16 and 17 , and so on. (The "t" then stands for leaves that are twos and threes, the " $f$ " for leaves of fours and fives, etc.) The mean diameter of the 102 ash trees in this sample is $11.500$ inches, and the standard deviation is $3.842$ inches. Linn County Trees in 1830 Ash Diameters $1 \mid 0 = 10$ inches $\quad \mathrm { N } = 102$ $0 . \mid$ ot $\mid 2$ of $\mid 44$ os $\mid 666777$ $0 * \mid 8888888888888888888999$ $1 . \mid 000000000000000000$ $1 t \mid 22222222222222222222222$ $1 f \mid 444444444445$ $1 s \mid 666666$ $1 * \mid 8888888888$ $2 \cdot \mid 0$ $2 t \mid$ $2 f \mid 4$ $2 s \mid$ $2 * \mid$ (a) What is the approximate diameter of an ash tree at the 20th percentile in this distribution? (b) According to the Empirical Rule, approximately 68% of ash tree diameters are between what two values?

The interquartile range is a measure of spread in a set of data.

Higher vertebrates (mammals and birds) exhibit lateralized behaviors, commonly referred to as "handedness." Recently an investigator studied the coiling behavior of the Agkistrodon piscivorus leucostoma - more commonly known as the cottonmouth snake. The cottonmouth is a venomous snake that spends much of its time in a coil, sometimes coiled in a counter-clockwise direction, other times in a clockwise direction. The investigator made many observations of coiling behavior, and calculated a "laterality index" for 30 cottonmouths. He defined the laterality index for each snake as the percentage of observations of the snake that were categorized as coils in the counter- clockwise direction. In the table below the data on laterality has been separated into two groups: females and males. Laterality: Females 58 55 61 54 64 60 42 49 56 42 59 55 55 60 66 70 49 51 57 43 Laterality: Males 56 56 52 54 40 49 49 48 46 49 -Construct comparative box plots of the laterality indices of female and male cottonmouths.

Higher vertebrates (mammals and birds) exhibit lateralized behaviors, commonly referred to as "handedness." Recently an investigator studied the coiling behavior of the Agkistrodon piscivorus leucostoma - more commonly known as the cottonmouth snake. The cottonmouth is a venomous snake that spends much of its time in a coil, sometimes coiled in a counter-clockwise direction, other times in a clockwise direction. The investigator made many observations of coiling behavior, and calculated a "laterality index" for 30 cottonmouths. He defined the laterality index for each snake as the percentage of observations of the snake that were categorized as coils in the counter- clockwise direction. In the table below the data on laterality has been separated into two groups: females and males. Laterality: Females 58 55 61 54 64 60 42 49 56 42 59 55 55 60 66 70 49 51 57 43 Laterality: Males 56 56 52 54 40 49 49 48 46 49 -Calculate the values of these numerical summaries for the laterality index for the sample of male cottonmouths: a) The mean __________ b) The standard deviation __________ The table below presents some summary statistics from the distribution of laterality index of males and females. Partial summary of data Laterality index Age Minimum Lower Quartile Median Upper Quartile Maximum Females 42 50 55.5 60 70 Males 40 48 49 54 56

The forces that determine the size of groups ("swarms") of social insects and the rates at which they grow are not well understood. Biologists have observed large variability in the size of swarms across species. In a study of the social wasp, Polybia occidentalis, investigators dismantled a nest of these insects and marked a few insects for future identification in new swarms. Twenty-five days after dismantling the original swarm they had located new swarms of wasps from the original colonies. The data below present the numbers of adults and workers in the new swarms in two different years. "Drones" are fertile males that mate with the queen. "Workers" are infertile females that labor in the nest and defend the colony. First Year Swarm \# Number of Drones Number of Workers Second Year Swarm \# Number of Drones Number of Workers 1 598 597 1 355 339 2 24 21 2 691 673 3 567 557 3 278 252 4 371 325 4 719 669 5 279 260 5 152 129 6 44 35 6 156 140 7 42 41 7 41 35 8 126 107 9 108 101 10 79 64 -Calculate the values of these numerical summaries for the number of drones in the second year swarms: a) The mean __________ b) The standard deviation ___________ The table below presents some summary statistics from the distributions of the number of drones. Partial summary of data Numbers of drones Swarm Minimum Lower Quartile Median Upper Quartile Maximum First year 24 43.5 117 420 598 Second year 41 152 278 691 719

One disadvantage of using the mean as a measure of center for a data set is that its value can be affected by the presence of even a single outlier in the data set.

Using appropriate statistical terminology, compare the center, shape, and spread of times on the walking task with and without the device.

Multiple sclerosis (MS) is an autoimmune disease that affects the brain and spinal cord. MS affects women more than men, and is most commonly diagnosed between the ages of 20 and 40. MS sufferers typically have problems of coordination and walking. The standard care for some sufferers is a plastic device affixed to the lower leg and ankle. In a recent test of this device, volunteers with MS were randomly assigned to two experimental groups. The volunteers in one group used the plastic device; the volunteers in the other group did not use the plastic device. Each individual was timed as he or she climbed a set of stairs. Times, measured without the device (ND) and with the device (D), are presented below. Stairs ND 9.3 8.4 8.6 6.9 17.7 12.4 18.6 14.2 7.9 6.9 8.4 9.4 12.7 Stairs D 7.9 8.2 9.0 7.2 15.3 13.5 16.6 15.7 7.7 7.5 8.4 9.5 12.7 -Using appropriate statistical terminology, compare the center, shape, and spread of times on the stair-climbing with and without the device.

Higher vertebrates (mammals and birds) exhibit lateralized behaviors, commonly referred to as "handedness." Recently an investigator studied the coiling behavior of the Agkistrodon piscivorus leucostoma - more commonly known as the cottonmouth snake. The cottonmouth is a venomous snake that spends much of its time in a coil, sometimes coiled in a counter-clockwise direction, other times in a clockwise direction. The investigator made many observations of coiling behavior, and calculated a "laterality index" for 30 cottonmouths. He defined the laterality index for each snake as the percentage of observations of the snake that were categorized as coils in the counter- clockwise direction. In the table below the data on laterality has been separated into two groups: females and males. Laterality: Females 58 55 61 54 64 60 42 49 56 42 59 55 55 60 66 70 49 51 57 43 Laterality: Males 56 56 52 54 40 49 49 48 46 49 -Grey Kangaroos are large social marsupials, indigenous to Australia. As part of a study of these creatures, biologists measured various aspects of their skeletal structure. A sample of 148 palate widths is presented in the stem-and-leaf plot below. The display uses five lines for each stem. Thus, "2t|" is the stem for palate widths of 22 and 23 , "2f|" for 24 and 25, "2s|" for 26 and 27, and so on. (The "t" then stands for leaves that are twos and threes, the " $f$ " for leaves of fours and fives, etc. " $2 . "$ Indicates leaves that are zeros and ones; " $2 *$ " indicates leaves that are eights and nines.) The mean palate width of this sample is $2.6 \mathrm {~cm}$ , and the standard deviation is $0.3 \mathrm {~cm}$ . Palate Width - Grey Kangaroos $1 \mid 0=1 \mathrm{~cm} \quad N=124$ 1. | 1t | 1f | 1s | 77 1* | 89 2. | 01111111111 2t | 222222222223333333333333 2 | 444444444444444455555555555555 2s | 6666666666667777777 2* | 88888888888889999999999999 3. | 10000011 3t | 223 3f | 3s | 3* | (a) Approximately what percent of palate widths in this sample are less than $2.9 \mathrm {~cm}$ ? (b) What is the approximate percentile of a palate width that is $2.0 \mathrm {~cm}$ ?

The Economist is a weekly newspaper that focuses on international politics, business news, and opinion. Each year the "Big Mac Index," the price (\$ U. S.) of a Big Mac in countries around the world, is published as a light-hearted way of comparing currencies. The values for 19 countries are displayed at right. ( $\$ 3.41$ was the U.S. price at publication in the summer of 2007.) BMI (USD) 3.41 1.68 2.67 4.17 2.95 1.54 3.61 2.29 4.01 1.60 3.68 2.69 1.45 2.51 2.51 2.03 3.53 2.59 5.08 (a) Construct a box plot for these data (b) Would you say this distribution is skewed or approximately symmetric? Justify your response using appropriate statistical terminology.

The % by weight of the element Lead was also calculated for Roman Mint: Lead Content (\% by weight) Statistic Value N 21 Mean 0.61 Median 0.62 St. Dev. 0.25 Minimum 0.09 Maximum 1.29 1st Quartile 0.41 3rd Quartile 0.73 present in the data. (b) Using your procedure from part (a), determine if there are outliers in these data.

Higher vertebrates (mammals and birds) exhibit lateralized behaviors, commonly referred to as "handedness." Recently an investigator studied the coiling behavior of the Agkistrodon piscivorus leucostoma - more commonly known as the cottonmouth snake. The cottonmouth is a venomous snake that spends much of its time in a coil, sometimes coiled in a counter-clockwise direction, other times in a clockwise direction. The investigator made many observations of coiling behavior, and calculated a "laterality index" for 30 cottonmouths. He defined the laterality index for each snake as the percentage of observations of the snake that were categorized as coils in the counter- clockwise direction. In the table below the data on laterality has been separated into two groups: females and males. Laterality: Females 58 55 61 54 64 60 42 49 56 42 59 55 55 60 66 70 49 51 57 43 Laterality: Males 56 56 52 54 40 49 49 48 46 49 -Using your box plots from part (a), compare the distributions of laterality index for males and females. Justify your comparisons by appealing to specific aspects of the box plots constructed in Question 2.

Does it appear from your graphs above that the device helps the MS sufferers in the walking test? Explain your reasoning in a few sentences.