Consider The Data Set:188, 190, 199, 181, 173, 192, 184, 197What Is The Standard Deviation Of The Data Set? Round Your Answer To The Nearest Tenth.Enter Your Answer In The Box.

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Understanding Standard Deviation

Standard deviation is a statistical measure that calculates the amount of variation or dispersion from the average of a set of values. It is a way to quantify the amount of spread or dispersion in a data set. In other words, it measures how much each value in the data set deviates from the mean value. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.

Calculating Standard Deviation

To calculate the standard deviation of a data set, we need to follow these steps:

  1. Find the mean: First, we need to find the mean of the data set. The mean is also known as the average. To find the mean, we add up all the values in the data set and divide by the number of values.

  2. Find the deviations: Next, we need to find the deviations from the mean. To do this, we subtract the mean from each value in the data set.

  3. Square the deviations: After finding the deviations, we need to square each deviation. This is done by multiplying each deviation by itself.

  4. Find the sum of the squared deviations: Next, we need to find the sum of the squared deviations. This is done by adding up all the squared deviations.

  5. Divide by the number of values minus one: To find the variance, we divide the sum of the squared deviations by the number of values minus one.

  6. Take the square root: Finally, we take the square root of the variance to find the standard deviation.

Calculating Standard Deviation for the Given Data Set

Now, let's apply these steps to the given data set: 188, 190, 199, 181, 173, 192, 184, 197.

Step 1: Find the Mean

To find the mean, we add up all the values in the data set and divide by the number of values.

Mean = (188 + 190 + 199 + 181 + 173 + 192 + 184 + 197) / 8
Mean = 1504 / 8
Mean = 188

Step 2: Find the Deviations

Next, we need to find the deviations from the mean. To do this, we subtract the mean from each value in the data set.

Deviations:
- 188 - 188 = 0
- 190 - 188 = 2
- 199 - 188 = 11
- 181 - 188 = -7
- 173 - 188 = -15
- 192 - 188 = 4
- 184 - 188 = -4
- 197 - 188 = 9

Step 3: Square the Deviations

After finding the deviations, we need to square each deviation.

Squared Deviations:
- 0^2 = 0
- 2^2 = 4
- 11^2 = 121
- (-7)^2 = 49
- (-15)^2 = 225
- 4^2 = 16
- (-4)^2 = 16
- 9^2 = 81

Step 4: Find the Sum of the Squared Deviations

Next, we need to find the sum of the squared deviations.

Sum of Squared Deviations = 0 + 4 + 121 + 49 + 225 + 16 + 16 + 81
Sum of Squared Deviations = 512

Step 5: Divide by the Number of Values Minus One

To find the variance, we divide the sum of the squared deviations by the number of values minus one.

Variance = 512 / (8 - 1)
Variance = 512 / 7
Variance = 73.14

Step 6: Take the Square Root

Finally, we take the square root of the variance to find the standard deviation.

Standard Deviation = √73.14
Standard Deviation = 8.55

Conclusion

In this article, we learned how to calculate the standard deviation of a data set. We applied the steps to the given data set: 188, 190, 199, 181, 173, 192, 184, 197. The standard deviation of the data set is 8.6 when rounded to the nearest tenth.

Final Answer

Understanding Standard Deviation

Standard deviation is a statistical measure that calculates the amount of variation or dispersion from the average of a set of values. It is a way to quantify the amount of spread or dispersion in a data set. In other words, it measures how much each value in the data set deviates from the mean value. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.

Frequently Asked Questions

Q1: What is the difference between standard deviation and variance?

A1: The variance is the average of the squared differences from the Mean. The standard deviation is the square root of the variance.

Q2: How do I calculate the standard deviation of a data set?

A2: To calculate the standard deviation of a data set, you need to follow these steps:

  1. Find the mean of the data set.
  2. Find the deviations from the mean.
  3. Square the deviations.
  4. Find the sum of the squared deviations.
  5. Divide the sum of the squared deviations by the number of values minus one.
  6. Take the square root of the result.

Q3: What is the formula for calculating the standard deviation?

A3: The formula for calculating the standard deviation is:

σ = √[(Σ(xi - μ)^2) / (n - 1)]

where σ is the standard deviation, xi is each value in the data set, μ is the mean, n is the number of values, and Σ denotes the sum.

Q4: What is the difference between population standard deviation and sample standard deviation?

A4: The population standard deviation is used when you have the entire population of data, while the sample standard deviation is used when you have a sample of the population.

Q5: How do I interpret the standard deviation of a data set?

A5: The standard deviation of a data set can be interpreted as follows:

  • A low standard deviation indicates that the values tend to be close to the mean.
  • A high standard deviation indicates that the values are spread out over a wider range.
  • A standard deviation of 0 indicates that all values are equal.

Q6: What is the relationship between standard deviation and the normal distribution?

A6: The standard deviation is related to the normal distribution in that the normal distribution is a probability distribution that is symmetric about the mean, and the standard deviation is a measure of the spread of the distribution.

Q7: Can I use the standard deviation to compare two data sets?

A7: Yes, you can use the standard deviation to compare two data sets. A lower standard deviation indicates that the values in the data set are more consistent, while a higher standard deviation indicates that the values are more variable.

Q8: What is the effect of outliers on the standard deviation?

A8: Outliers can significantly affect the standard deviation of a data set. If a data set contains outliers, the standard deviation will be higher than it would be if the outliers were not present.

Q9: Can I use the standard deviation to predict future values?

A9: No, the standard deviation is not a reliable method for predicting future values. It is a measure of the spread of a data set, and it does not take into account the underlying trends or patterns in the data.

Q10: What are some common applications of the standard deviation?

A10: The standard deviation has many applications in statistics and data analysis, including:

  • Describing the spread of a data set
  • Comparing the spread of two or more data sets
  • Identifying outliers in a data set
  • Predicting future values (although this is not a reliable method)

Conclusion

In this article, we answered some frequently asked questions about standard deviation. We covered topics such as calculating the standard deviation, interpreting the standard deviation, and using the standard deviation to compare data sets. We also discussed the relationship between standard deviation and the normal distribution, and the effect of outliers on the standard deviation.