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Chapter 3: Problem 7

Find the mean, median, and mode of the data set 8 2 7 2 6 5

Short Answer

Expert verified
Mean: 5, Median: 5.5, Mode: 2

Step by step solution

01

Arrange the Data

First, arrange the numbers in the data set in ascending order: 2, 2, 5, 6, 7, 8.
02

Calculate the Mean

The mean is the average of the numbers. To find it, add all the numbers together and divide by the total number of numbers. \[\text{Mean} = \frac{2 + 2 + 5 + 6 + 7 + 8}{6} = \frac{30}{6} = 5.\]
03

Determine the Median

The median is the middle number when the data is ordered. Since there are 6 numbers (an even count), the median will be the average of the third and fourth numbers in the ordered list: \[\text{Median} = \frac{5 + 6}{2} = \frac{11}{2} = 5.5.\]
04

Identify the Mode

The mode is the number that appears most frequently in the data set. The number 2 appears twice, whereas all other numbers appear only once. Therefore, the mode is 2.

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Key Concepts

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

Understanding Mean Calculation
The mean, commonly known as the average, helps us understand the central tendency of a data set. It gives us an idea of where the center of a data collection lies. To calculate the mean:
  • Add up all the numbers in the data set.
  • Divide the total by the number of data points (in this case, 6).
Applying this to our data set: 2, 2, 5, 6, 7, 8, we sum them to get 30. Dividing by 6 yields a mean of 5. This number tells us that if the total value of the data were spread evenly across all observations, each one would have a value of 5. It helps in analyzing the overall behavior of data, although it's sensitive to extreme values, or outliers.
Grasping Median Determination
The median is another measure of central tendency that divides a data set into two equal halves. It identifies the middle value when numbers are arranged in ascending or descending order. This middle value is often useful for understanding the distribution's center without getting influenced heavily by outliers.
In our data set example: 2, 2, 5, 6, 7, 8, since there are 6 numbers (an even set), the median is the average of the third and fourth values. Looking at 5 and 6, we calculate the median as:
\[\text{Median} = \frac{5 + 6}{2} = 5.5.\]The median value of 5.5 provides a balance point, reflecting the midpoint of the ordered set and giving us insight into the data's spread.
Exploring Mode Identification
The mode is the number that occurs most frequently in a data set. It's a straightforward concept, as it simply shows the value with the most repetitions.
In our example data: 2, 2, 5, 6, 7, 8, the number 2 appears twice, more than any other number. Therefore, the mode is identified as 2.
  • If no number repeats, the dataset might have no mode.
  • If multiple numbers repeat with the same highest frequency, the dataset could be multimodal.
Knowing the mode helps in understanding the most common occurrence in your data set, providing another angle to observe the dataset's characteristics.

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Most popular questions from this chapter

Critical Thinking Consider a data set of 15 distinct measurements with mean \(A\) and median \(B\) (a) If the highest number were increased, what would be the effect on the median and mean? Explain. (b) If the highest number were decreased to a value still larger than \(B\), what would be the effect on the median and mean? (c) If the highest number were decreased to a value smaller than \(B\), what would be the effect on the median and mean?

How hot does it get in Death Valley? The following data are taken from a study conducted by the National Park System, of which Death Valley is a unit. The ground temperatures ( \(^{\circ} \mathrm{F}\) ) were taken from May to November in the vicinity of Furnace Creek. $$\begin{array}{ccccccc}146 & 152 & 168 & 174 & 180 & 178 & 179 \\\180 & 178 & 178 & 168 & 165 & 152 & 144\end{array}$$ Compute the mean, median, and mode for these ground temperatures.

Consider two data sets. Set A: \(n=5 ; \bar{x}=10 \quad\) Set \(\mathrm{B}: n=50 ; \bar{x}=10\) (a) Suppose the number 20 is included as an additional data value in Set A. Compute \(\bar{x}\) for the new data set. Hint: \(\Sigma x=n \bar{x} .\) To compute \(\bar{x}\) for the new data set, add 20 to \(\Sigma x\) of the original data set and divide by 6. (b) Suppose the number 20 is included as an additional data value in Set B. Compute \(\bar{x}\) for the new data set. (c) Why does the addition of the number 20 to each data set change the mean for Set A more than it does for Set B?

For a given data set in which not all data values are equal, which value is smaller, \(s\) or \(\sigma ?\) Explain.

In this problem, we explore the effect on the mean, median, and mode of adding the same number to each data value. Consider the data set 2,2,3,6,10. (a) Compute the mode, median, and mean. (b) Add 5 to each of the data values. Compute the mode, median, and mean. (c) Compare the results of parts (a) and (b). In general, how do you think the mode, median, and mean are affected when the same constant is added to each data value in a set?
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