Given the following data:
5, 7, 10, 12, 15, 17
* Find the mean deviation from the mean.
* Find the mean deviation from the median.
Answer (2)
Calculate the mean of the data set: mean = 6 5 + 7 + 10 + 12 + 15 + 17 = 11 .
Calculate the median of the data set: median = 2 10 + 12 = 11 .
Calculate the mean deviation from the mean: MD mean = 6 ∣5 − 11∣ + ∣7 − 11∣ + ∣10 − 11∣ + ∣12 − 11∣ + ∣15 − 11∣ + ∣17 − 11∣ = 6 22 = 3 11 ≈ 3.67 .
Calculate the mean deviation from the median: MD median = 6 ∣5 − 11∣ + ∣7 − 11∣ + ∣10 − 11∣ + ∣12 − 11∣ + ∣15 − 11∣ + ∣17 − 11∣ = 6 22 = 3 11 ≈ 3.67 .
The mean deviation from the mean and the mean deviation from the median are both 3 11 .
Explanation
Understanding the Problem We are given the data set 5, 7, 10, 12, 15, 17 and asked to find the mean deviation from the mean and the mean deviation from the median.
Calculating the Mean First, we need to calculate the mean of the data set. The mean is the sum of all the data points divided by the number of data points. In this case, the mean is calculated as follows: mean = 6 5 + 7 + 10 + 12 + 15 + 17 = 6 66 = 11
Calculating the Median Next, we calculate the median of the data set. Since there are 6 data points (an even number), the median is the average of the middle two numbers. The middle two numbers are 10 and 12. Thus, the median is calculated as follows: median = 2 10 + 12 = 2 22 = 11
Calculating Mean Deviation from the Mean Now, we calculate the mean deviation from the mean. The mean deviation from the mean is the average of the absolute differences between each data point and the mean. In this case, the mean deviation from the mean is calculated as follows: MD mean = 6 ∣5 − 11∣ + ∣7 − 11∣ + ∣10 − 11∣ + ∣12 − 11∣ + ∣15 − 11∣ + ∣17 − 11∣ MD mean = 6 ∣ − 6∣ + ∣ − 4∣ + ∣ − 1∣ + ∣1∣ + ∣4∣ + ∣6∣ = 6 6 + 4 + 1 + 1 + 4 + 6 = 6 22 = 3 11 ≈ 3.67
Calculating Mean Deviation from the Median Finally, we calculate the mean deviation from the median. The mean deviation from the median is the average of the absolute differences between each data point and the median. In this case, the mean deviation from the median is calculated as follows: MD median = 6 ∣5 − 11∣ + ∣7 − 11∣ + ∣10 − 11∣ + ∣12 − 11∣ + ∣15 − 11∣ + ∣17 − 11∣ MD median = 6 ∣ − 6∣ + ∣ − 4∣ + ∣ − 1∣ + ∣1∣ + ∣4∣ + ∣6∣ = 6 6 + 4 + 1 + 1 + 4 + 6 = 6 22 = 3 11 ≈ 3.67
Final Answer Therefore, the mean deviation from the mean is 3 11 ≈ 3.67 and the mean deviation from the median is 3 11 ≈ 3.67 .
Examples
Understanding mean and median deviations helps in analyzing the spread of data, which is crucial in fields like finance. For example, when assessing investment risk, knowing how much individual returns deviate from the average (mean) or the middle value (median) provides insights into the stability and predictability of the investment. Lower deviations indicate more consistent returns, while higher deviations suggest greater volatility.
The mean deviation from the mean for the data set {5, 7, 10, 12, 15, 17} is approximately 3.67, and the mean deviation from the median is also approximately 3.67. Both calculations yield the same result in this case.
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