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Chapter 3: Problem 10

During their first swim through a water maze, 15 laboratory rats made the following number of errors (blind alleyway entrances): 2,17,5,3,28,7,5,8,5,6,2,12,10,4,3 . (a) Find the mode, median, and mean for these data. (b) Without constructing a frequency distribution or graph, would you characterize the shape of this distribution as balanced, positively skewed, or negatively skewed?

Short Answer

Expert verified
The mode is 5, the median is 5, and the mean is 7.93. The distribution of data is positively skewed.

Step by step solution

01

Calculating the Mean

Firstly, the mean (average) for these data can be found. The mean is the sum of all values divided by the number of values. In this case, the sum of the errors is 119, and there are 15 rats, so the mean is \( \frac{119}{15} \) = 7.93.
02

Calculating the Median

Next, arrange the data in ascending order, which will aid in finding the median. The sorted data is: 2, 2, 3, 3, 4, 5, 5, 5, 6, 7, 8, 10, 12, 17, 28. The median is the middle value in the sorted list. Since there are 15 values, the median is the 8th value, which is 5.
03

Calculating the Mode

The mode is the value(s) that appears most frequently in the data. Here, the value 5 appears three times, more than any other value, so the mode is 5.
04

Determining skewness

To determine the skewness without constructing a frequency distribution or graph, compare the mean and the median. If the mean is greater than the median, the distribution is positively skewed, if smaller - negatively skewed, and if they are about equal - the distribution is balanced. Here, the mean (7.93) is greater than the median (5), so the distribution of data is positively skewed.

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Key Concepts

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

Understanding the Mean
The mean is often the first statistic people think of when summarizing data. It's a measure of central tendency, showing where the "center" of a data set is. To find the mean, simply add all the numbers in the data set together. Then, divide by how many numbers there are.

In our example, the errors made by 15 rats add up to 119. Since there are 15 values, the mean is calculated as follows: \[ \text{Mean} = \frac{119}{15} = 7.93 \] This value represents the average number of errors made by the rats.

The mean gives a quick snapshot but can be influenced by extreme values. For instance, a single rat with a high error count of 28 affects the mean significantly.
Decoding the Median
While the mean can be skewed by outliers, the median offers a different story. The median represents the middle value when data is sorted in order.

Let's look at our sorted list of errors: 2, 2, 3, 3, 4, 5, 5, 5, 6, 7, 8, 10, 12, 17, 28. With 15 values, the middle one is the 8th, which is 5. So, the median is 5.

The median is often more robust against outliers than the mean. This means it doesn't get dragged around by extremely high or low values. This makes it a reliable measure of central tendency, especially for skewed distributions.
Understanding the Mode
The mode is perhaps the simplest to understand. It's the number that appears most frequently in your data set. You might even spot it without doing any calculations.

In the list of rat errors, the number 5 occurs three times, more than any other number. Thus, the mode is 5.

The mode can tell us what is common or typical for our data. However, it's possible to have more than one mode (if multiple so occur equally often), or even no mode at all if no number repeats. In our case, having a clear mode indicates a common error level among the rats.
Determining the Skewness
Skewness is about the shape of our data distribution. It tells us how data is spread, compared to normal distribution.
  • Positively skewed: when the mean is greater than the median.
  • Negatively skewed: when the mean is less than the median.
  • Balanced: when the mean and median are nearly equal.
In our rat example, the mean is 7.93 and the median is 5. Since the mean is higher, the distribution is positively skewed. This suggests that there are a few rats with high error counts pulling the average up.

Understanding skewness helps predict data tendencies. For instance, in a positively skewed distribution like ours, most rats tended to make fewer errors, but a few made significantly more.

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Most popular questions from this chapter

The mean serves as the balance point for any distribution because the sum of all scores, expressed as positive and negative distances from the mean, always equals zero. (a) Show that the mean possesses this property for the following set of scores: \(3,6,2,\) 0,4 (b) Satisfy yourself that the mean identifies the only point that possesses this property. More specifically, select some other number, preferably a whole number (for convenience), and then find the sum of all scores in part (a), expressed as positive or negative distances from the newly selected number. This sum should not equal zero.

Indicate whether each of the following distributions is positively or negatively skewed. The distribution of (a) incomes of taxpayers has a mean of \(\$ 48,000\) and a median of \(\$ 43,000\). (b) GPAs for all students at some college has a mean of 3.01 and a median of 3.20 . (c) number of "romantic affairs" reported anonymously by young adults has a mean of 2.6 affairs and a median of 1.9 affairs. (d) daily TV viewing times for preschool children has a mean of 55 minutes and a median of 73 minutes.

Find the mean for the following retirement ages: \(60,63,45,63,65,\) \(70,55,63,60,65,63 .\)

Indicate whether the following terms or symbols are associated with the population mean, the sample mean, or both means. (a) \(N\) (b) varies (c) \(\sum\) (c) \(\mathrm{n}\) (d) constant (e) subset

In some racing events, downhill skiers receive the average of their times for three trials. Would you prefer the average time to be the mean or the median if usually you have (a) one very poor time and two average times? (b) one very good time and two average times? (c) two good times and one average time? (d) three different times, spaced at about equal intervals?
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