C.1 Check for Understanding
Activity C.1.1 Identify and compute the appropriate measures of central tendency using the step-by-step process of the following. Please include to answer the measures of dispersion (Range, Variance and Standard Deviation) of items

1. Look at the list of shoe sizes worn by the basketball team members. Which shoe size appears most frequently among the player. How might knowing the most common shoe size help when ordering team shoes or ensuring availability for everyone. (Shoe sizes in basketball Team: 8, 9, 9, 10, 10, 10, 11, 12)

To tackle the given problem, we need to calculate the measures of central tendency (mean, median, mode) and the measures of dispersion (range, variance, and standard deviation) for the basketball team's shoe sizes.

### **Step 1: Organize the Data**
Shoe sizes: 
\[ 8, 9, 9, 10, 10, 10, 11, 12 \]

### **Step 2: Calculate the Measures of Central Tendency**

#### **Mode**:
The mode is the value that appears most frequently in the data set.

- Shoe sizes: 8, 9, 9, 10, 10, 10, 11, 12
- The mode is **10** because it appears three times, which is more frequent than any other value.

#### **Median**:
The median is the middle value of a data set when it is ordered from least to greatest. If there is an even number of data points, the median is the average of the two middle values.

- Ordered shoe sizes: 8, 9, 9, 10, 10, 10, 11, 12
- The two middle values are the 4th and 5th data points, both of which are **10**.
- So, the median is **10**.

#### **Mean**:
The mean (average) is the sum of all values divided by the number of values.

\[
\text{Mean} = \frac{8 + 9 + 9 + 10 + 10 + 10 + 11 + 12}{8} = \frac{79}{8} = 9.875
\]
So, the mean shoe size is **9.875**.

### **Step 3: Calculate the Measures of Dispersion**

#### **Range**:
The range is the difference between the maximum and minimum values in the data set.

\[
\text{Range} = 12 - 8 = 4
\]
So, the range is **4**.

#### **Variance**:
Variance measures how far the data points are from the mean. To compute variance, follow these steps:

1. Subtract the mean from each data point.
2. Square the result for each data point.
3. Sum these squared differences.
4. Divide by the number of data points (for population variance).

| Shoe Size | Deviation from Mean (x - 9.875) | Squared Deviation |
|-----------|----------------------------------|-------------------|
| 8         | 8 - 9.875 = -1.875               | 3.5156            |
| 9         | 9 - 9.875 = -0.875               | 0.7656            |
| 9         | 9 - 9.875 = -0.875               | 0.7656            |
| 10        | 10 - 9.875 = 0.125               | 0.0156            |
| 10        | 10 - 9.875 = 0.125               | 0.0156            |
| 10        | 10 - 9.875 = 0.125               | 0.0156            |
| 11        | 11 - 9.875 = 1.125               | 1.2656            |
| 12        | 12 - 9.875 = 2.125               | 4.5156            |

- Sum of squared deviations: \( 3.5156 + 0.7656 + 0.7656 + 0.0156 + 0.0156 + 0.0156 + 1.2656 + 4.5156 = 10.8752 \)
- Population variance: \( \frac{10.8752}{8} = 1.3594 \)

So, the variance is **1.3594**.

#### **Standard Deviation**:
The standard deviation is the square root of the variance.

\[
\text{Standard Deviation} = \sqrt{1.3594} \approx 1.166
\]

So, the standard deviation is approximately **1.166**.

### **Step 4: Interpretation of Mode**
Knowing that the mode of the shoe sizes is **10**, we understand that most players wear size 10 shoes. This information can be helpful when ordering shoes because it ensures the availability of the most common shoe size, reducing the risk of players not having the correct size.

---

### **Summary of Results**:
- **Mode**: 10
- **Median**: 10
- **Mean**: 9.875
- **Range**: 4
- **Variance**: 1.3594
- **Standard Deviation**: 1.166

---

Would you like any further clarification or additional details?

### Related Questions:
1. How would the results change if a new shoe size, 13, is added?
2. What impact does a large range have on the interpretation of the data?
3. How is the variance used in other statistical contexts?
4. Can there be more than one mode in a dataset?
5. Why is the mean sometimes not the best measure of central tendency?

### Tip:
When working with small datasets, the mode can be particularly useful for understanding common values.

C.1 Check for Understanding Activity C.1.1 Identify and compute the appropriate measures of central tendency using the step-by-step process of the following. Please include to answer the measures of dispersion (Range, Variance and Standard Deviation) of items

1. Look at the list of shoe sizes worn by the basketball team members. Which shoe size appears most frequently among the player. How might knowing the most common shoe size help when ordering team shoes or ensuring availability for everyone. (Shoe sizes in basketball Team: 8, 9, 9, 10, 10, 10, 11, 12)

Learn how to calculate the central tendency (mean, median, mode) and measures of dispersion (range, variance, standard deviation) for a basketball team's shoe sizes. Shoe sizes: 8, 9, 9, 10, 10, 10, 11, 12. This problem demonstrates key statistical concepts and is ideal for students in Grades 9-12.

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