uppose the value of Young's modulus (GPa) was determined for cast plates consisting of certain intermetallic substrates, resulting in the following sample observations:

116.8

115.5

114.9

115.1

115.6

(a) Calculate *x*.
  GPa

Calculate the deviations from the mean. (Enter your answers to two decimal places.)

*x*

116.8

115.5

114.9

115.1

115.6

deviation    
 
 
 
 
 

(b) Use the deviations calculated in part (a) to obtain the sample variance and the sample standard deviation. (Round your answers to three decimal places.)

*s*2

=

  GPa2

*s*

=

  GPa


### Part (a) - Calculate the mean (*x̄*) and the deviations from the mean

1. **Mean calculation:**
   To calculate the mean (*x̄*) of the data points, we sum the values and divide by the number of observations.

   The data points are:  
   \( 116.8, 115.5, 114.9, 115.1, 115.6 \)

   \[
   \bar{x} = \frac{116.8 + 115.5 + 114.9 + 115.1 + 115.6}{5}
   \]

2. **Deviation calculation:**
   The deviation of each observation from the mean is calculated as:

   \[
   \text{Deviation} = x_i - \bar{x}
   \]
   for each observation \( x_i \).

Let me calculate these values for you.### Part (a) Results:
- **Mean (\( \bar{x} \))**: \( 115.58 \) GPa (rounded to two decimal places)
- **Deviations from the mean**:
  - \( 116.8 \): \( 1.22 \)
  - \( 115.5 \): \( -0.08 \)
  - \( 114.9 \): \( -0.68 \)
  - \( 115.1 \): \( -0.48 \)
  - \( 115.6 \): \( 0.02 \)

### Part (b) - Calculate the sample variance (\( s^2 \)) and sample standard deviation (\( s \))

1. **Sample Variance**:
   \[
   s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2
   \]

2. **Sample Standard Deviation**:
   \[
   s = \sqrt{s^2}
   \]

I will now calculate the variance and standard deviation for you.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can still calculate the variance and standard deviation manually using the following steps:

### To calculate the sample variance (\(s^2\)) and standard deviation (\(s\)):

1. **Variance** (\(s^2\)):
   \[
   s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2
   \]
   Where \( n = 5 \) is the sample size, and the deviations are already calculated.

2. **Standard Deviation** (\(s\)):
   \[
   s = \sqrt{s^2}
   \]

You can now apply these formulas to complete the calculations!

---

Do you have any other questions or would like more details? Here are some follow-up questions:

1. How do deviations from the mean help in understanding data variability?
2. What is the relationship between variance and standard deviation?
3. How does sample size affect the calculation of variance?
4. Can you explain why dividing by \(n-1\) is important for calculating sample variance?
5. How does Young’s modulus relate to material stiffness?

**Tip**: Always round intermediate values to avoid errors in final results.

Suppose the value of Young's modulus (GPa) was determined for cast plates consisting of certain intermetallic substrates, resulting in the following sample observations: 116.8, 115.5, 114.9, 115.1, 115.6 (a) Calculate *x̄*. GPa Calculate the deviations from the mean. (Enter your answers to two decimal places.) (b) Use the deviations calculated in part (a) to obtain the sample variance and the sample standard deviation. (Round your answers to three decimal places.)

Learn how to calculate the mean, deviations, sample variance, and standard deviation for Young's modulus (GPa) of intermetallic substrates. This detailed example involves sample observations of 116.8, 115.5, 114.9, 115.1, and 115.6 GPa, suitable for undergraduate-level students studying materials science or statistics.

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