Find the standard deviation:

-2.7, 1.6, -5.2, -3.7, -2.5, -3.3, -3.7, -2.1

To find the standard deviation of the data set \(-2.7, 1.6, -5.2, -3.7, -2.5, -3.3, -3.7, -2.1\), follow these steps:

### Step 1: Calculate the Mean
First, find the mean (average) of the data set.

\[
\text{Mean} = \frac{-2.7 + 1.6 + (-5.2) + (-3.7) + (-2.5) + (-3.3) + (-3.7) + (-2.1)}{8}
\]

\[
\text{Mean} = \frac{-21.6}{8} = -2.7
\]

### Step 2: Calculate the Squared Differences from the Mean
Next, subtract the mean from each number and square the result:

\[
\begin{align*}
(-2.7 - (-2.7))^2 & = (0.0)^2 = 0.0 \\
(1.6 - (-2.7))^2 & = (4.3)^2 = 18.49 \\
(-5.2 - (-2.7))^2 & = (-2.5)^2 = 6.25 \\
(-3.7 - (-2.7))^2 & = (-1.0)^2 = 1.0 \\
(-2.5 - (-2.7))^2 & = (0.2)^2 = 0.04 \\
(-3.3 - (-2.7))^2 & = (-0.6)^2 = 0.36 \\
(-3.7 - (-2.7))^2 & = (-1.0)^2 = 1.0 \\
(-2.1 - (-2.7))^2 & = (0.6)^2 = 0.36 \\
\end{align*}
\]

### Step 3: Find the Mean of the Squared Differences
Sum the squared differences and divide by the number of data points (since it's a sample, we use \(n-1\)):

\[
\text{Variance} = \frac{0.0 + 18.49 + 6.25 + 1.0 + 0.04 + 0.36 + 1.0 + 0.36}{8 - 1}
\]

\[
\text{Variance} = \frac{27.5}{7} \approx 3.9286
\]

### Step 4: Take the Square Root of the Variance
Finally, the standard deviation is the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{3.9286} \approx 1.982
\]

### Answer
The standard deviation of the data set is approximately **1.98**.

---

Would you like to delve deeper into any of these steps or ask additional questions?

Here are 5 related questions to consider:
1. How does changing the mean affect the standard deviation?
2. What is the difference between population standard deviation and sample standard deviation?
3. How would the standard deviation change if one of the values was doubled?
4. What is the impact of outliers on the standard deviation?
5. How does standard deviation relate to the concept of variance?

**Tip:** When analyzing data, the standard deviation helps to understand how much the data deviates from the average, providing a measure of consistency or variability.

Learn how to calculate the standard deviation for the data set -2.7, 1.6, -5.2, -3.7, -2.5, -3.3, -3.7, -2.1 with a step-by-step solution. Understand the concepts of mean, variance, and standard deviation. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation