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The paper referenced in the previous exercise also gave data on the actual amount (in \(\mathrm{ml}\) ) poured into a short, wide glass for individuals asked to pour 1.5 ounces \((44.3 \mathrm{ml})\) $$ \begin{array}{llllllll} 89.2 & 68.6 & 32.7 & 37.4 & 39.6 & 46.8 & 66.1 & 79.2 \\ 66.3 & 52.1 & 47.3 & 64.4 & 53.7 & 63.2 & 46.4 & 63.0 \\ 92.4 & 57.8 & & & & & & \end{array} $$

Short Answer

Expert verified
Organize the data in ascending order: \(32.7, 37.4, 39.6, 46.4, 46.8, 47.3, 52.1, 53.7, 57.8, 63.0, 63.2, 64.4, 66.1, 66.3, 68.6, 79.2, 89.2, 92.4\). Calculate the mean: \(\frac{32.7+37.4+\cdots+89.2+92.4}{18} \approx 59.53 \: ml\). Calculate the median: The middle values are \(57.8\) and \(63.0\), so the median is \(\frac{57.8+63.0}{2} = 60.4 \: ml\). Calculate the standard deviation: First, find the squared differences: \((32.7-59.53)^2, (37.4-59.53)^2, \cdots, (89.2-59.53)^2, (92.4-59.53)^2\). Find the average of these squared differences: \(\frac{726.43+487.89+\cdots+1078.25+1084.22}{18} \approx 337.27\), and take the square root to obtain the standard deviation, which is approximately \(18.36 \: ml\). The mean of the dataset is \(59.53 \: ml\), the median is \(60.4 \: ml\), and the standard deviation is \(18.36 \: ml\). This indicates that the individuals generally poured more than the desired \(44.3 \: ml\), and there was considerable variability in the amounts poured.

Step by step solution

01

Organize the data

Start by organizing the data given in the exercise in ascending order. This will make it easier to calculate the mean, median, and standard deviation.
02

Calculate mean

The mean (average) is the sum of all measurements divided by the number of measurements. Using the organized data, calculate the sum and divide by the total number of values to find the mean.
03

Calculate median

The median is the middle value of the data set. If the number of values is even, the median is the average of the middle two values. Locate the middle value(s) in your organized data and calculate the median.
04

Calculate standard deviation

The standard deviation measures the dispersion of the data set. First, calculate the mean. Then, for each value in the dataset, subtract the mean and square the result. Find the average of those squared differences, and take the square root of the average to obtain the standard deviation.
05

Interpret the results

Once you have calculated the mean, median, and standard deviation of the dataset, analyze the results to better understand the pattern in the data. The mean and median will give an indication of the central tendency of the dataset, while the standard deviation will show the dispersion of the values around that central tendency.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation
Calculating the mean is a straightforward process. It's essentially finding the average value of a dataset. To calculate the mean of a dataset, you first sum all the individual data points.
Next, you divide this sum by the number of data points you have.
For instance, let's say you have 5 numbers: 3, 7, 8, 6, and 5. To find the mean:
  • Add the numbers: 3 + 7 + 8 + 6 + 5 = 29
  • Divide by the count of numbers: 29 / 5 = 5.8
So, the mean is 5.8. In context, the mean provides us with a general indicator of the central value of the dataset. It can be especially useful for interpreting large datasets or when comparing different sets of data. However, be mindful that an extremely high or low value can skew the mean significantly.
Median Calculation
The median is another measure of central tendency, which differs from the mean. To find the median, you must first arrange your data in ascending order. Once the data is ordered, locating the median depends on whether the number of your data points is odd or even.
  • If the dataset has an odd number of observations, the median is the middle number.
  • If the dataset is even, the median is the average of the two middle numbers.
For example, consider the dataset: 9, 5, 6, 3, 8. First, order the data to 3, 5, 6, 8, 9. There are 5 data points, so the middle number (3rd value) is 6. Thus, the median is 6.
If you add another value, say 10, making the dataset 3, 5, 6, 8, 9, 10, then you average the two middle numbers (6 and 8). So, the median becomes 7.
Standard Deviation Calculation
The standard deviation gives us insight into how spread out the numbers in a dataset are.
To calculate the standard deviation, first calculate the mean of your dataset.
Then, subtract the mean from each number to find the deviation of each value from the mean.
  • Square each of these deviations to eliminate negative values.
  • Find the average of these squared deviations.
  • Finally, take the square root of this average to get the standard deviation.
This process might seem a bit involved, but it provides a powerful tool for understanding variability. A small standard deviation means the data points are close to the mean, while a large standard deviation indicates a wide spread of values.
Data Organization
Organizing data is a foundational step in any statistical analysis.
Properly organized data helps make calculations like mean, median, and standard deviation more straightforward.
Begin by arranging your dataset in either ascending or descending order.
This simple step helps in effortlessly identifying key statistics like the median or ranges.
  • With organized data, spotting errors or anomalies becomes simpler.
  • It enables better visualization, which aids in the comprehension of data trends and patterns.
Take the time to organize your data properly. Not only is it essential for calculating specific statistics, but it also provides clarity and insight into what the data represents. This step sets the foundation for accurate data analysis and ensures your conclusions are reliable.

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Most popular questions from this chapter

The accompanying data on total amount of time per day (in minutes) spent using a cell phone are consistent with summary statistics in the paper "The Relationship Between Cell Phone Use and Academic Performance in a Sample of U.S. College Students" (SAGE Open [2015]: 1-9). $$ \begin{array}{rrrrrrrrr} 225 & 318 & 468 & 0 & 236 & 601 & 144 & 196 & 374 \\ 0 & 424 & 198 & 156 & 734 & 331 & 502 & 0 & 492 \\ 563 & 195 & 237 & 110 & 516 & 422 & 740 & & \end{array} $$ Calculate and interpret the values of the median and the interquartile range.

The paper "Study of the Flying Ability of Rhynchophorus ferrugineus Adults Using a Computer-Monitored Mill" (Bulletin of Entomological Research [2014]: \(462-467\) ) summarized data from a study of red palm weevils, a pest that is a threat to palm trees. The following frequency distribution from the paper was constructed using the longest flight (in meters) observed for 132 weevils. $$ \begin{array}{|lc|} \hline \text { Longest Flight (m) } & \text { Frequency (Number of Weevils) } \\\ \hline 0 \text { to }<100 & 71 \\ 100 \text { to }<2,000 & 32 \\ 2000 \text { to }<5,000 & 18 \\ 5000 \text { to }<10,000 & 8 \\ 10,000 \text { or more } & 3 \\ \hline \end{array} $$ Estimate the approximate values of the following percentiles: a. 54 th b. 80 th c. \(92 \mathrm{nd}\)

The mean number of text messages sent per month by customers of a cell phone service provider is \(1650,\) and the standard deviation is 750 . Find the z-score associated with each of the following numbers of text messages sent. a. 0 b. 10,000 c. 4500 d. 300

Data on manufacturing defects per 100 cars for the 30 brands of cars sold in the United States (USA TODAY, March 29,2016\()\) are: $$ \begin{array}{lllllllllll} 97 & 134 & 198 & 142 & 95 & 135 & 132 & 145 & 136 & 129 & 152 \\ 158 & 169 & 155 & 106 & 125 & 120 & 153 & 208 & 163 & 204 & 173 \\ 165 & 126 & 113 & 167 & 171 & 166 & 181 & 161 & & & \end{array} $$

A student took two national aptitude tests. The mean and standard deviation were 475 and 100 , respectively, fol the first test, and 30 and 8 , respectively, for the second test The student scored 625 on the first test and 45 on the second test. Use \(z\) -scores to determine on which exam the student performed better relative to the other test takers. (Hint: See Example \(3.18 .)\)

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