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 solve this, we will find both the **measures of central tendency** (mean, median, mode) and the **measures of dispersion** (range, variance, and standard deviation) for the given shoe sizes in the basketball team. 

### Given dataTo solve this, we will find both the **measures of central tendency** (mean, median, mode) and the **measures of dispersion** (range, variance, and standard deviation) for the given shoe sizes in the basketball team. 

### Given 
Shoe sizes: 8, 9, 9, 10, 10, 10, 11, 12

---

### Step 1: Measures of Central Tendency

#### 1. **Mode**:
The mode is the value that appears most frequently in the dataset.

- Shoe sizes: 8, 9, 9, 10, 10, 10, 11, 12
- The most frequent shoe size is **10** (appears 3 times).

So, the **mode** is **10**.

Knowing the mode is helpful because it tells the coach which shoe size should be prioritized for ordering. Ensuring that shoe size 10 is always available would cover the majority of players.

#### 2. **Mean** (Average):
To find the mean, sum all the shoe sizes and divide by the number of players.

\[
\text{Mean} = \frac{8 + 9 + 9 + 10 + 10 + 10 + 11 + 12}{8}
\]

\[
\text{Mean} = \frac{79}{8} = 9.875
\]

So, the **mean** shoe size is approximately **9.88**.

#### 3. **Median**:
The median is the middle value when the data is arranged in ascending order.

- Ordered shoe sizes: 8, 9, 9, 10, 10, 10, 11, 12
- Since there are 8 players (an even number), the median is the average of the 4th and 5th values in the ordered list.
  - The 4th and 5th values are both **10**.

So, the **median** shoe size is **10**.

---

### Step 2: Measures of Dispersion

#### 1. **Range**:
The range is the difference between the largest and smallest values in the dataset.

\[
\text{Range} = 12 - 8 = 4
\]

So, the **range** is **4**.

#### 2. **Variance**:
The variance measures the spread of the data points. To find the variance, we follow these steps:

1. Calculate the mean: **9.875**
2. Subtract the mean from each shoe size and square the result:
   - (8 - 9.875)² = 3.515625
   - (9 - 9.875)² = 0.765625
   - (9 - 9.875)² = 0.765625
   - (10 - 9.875)² = 0.015625
   - (10 - 9.875)² = 0.015625
   - (10 - 9.875)² = 0.015625
   - (11 - 9.875)² = 1.265625
   - (12 - 9.875)² = 4.515625
3. Find the average of these squared differences:

\[
\text{Variance} = \frac{3.515625 + 0.765625 + 0.765625 + 0.015625 + 0.015625 + 0.015625 + 1.265625 + 4.515625}{8}
\]

\[
\text{Variance} = \frac{10.875}{8} = 1.359375
\]

So, the **variance** is **1.36** (rounded to 2 decimal places).

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

\[
\text{Standard Deviation} = \sqrt{1.359375} \approx 1.166
\]

So, the **standard deviation** is approximately **1.17**.

---

### Summary of Results:

- **Mode**: 10
- **Mean**: 9.88
- **Median**: 10
- **Range**: 4
- **Variance**: 1.36
- **Standard Deviation**: 1.17

---

### Application:
- **Mode (10)** is useful for ordering shoes because it tells which shoe size is most common, ensuring that more shoes of size 10 are available.
  
Would you like a further breakdown of any step or a different analysis?

---

**Related Questions:**
1. How would the measures of central tendency change if one player's shoe size was 13?
2. Can the variance and standard deviation provide insights into how diverse the shoe sizes are on the team?
3. What happens to the median and mode if the dataset is altered by adding more players with size 10 shoes?
4. How does knowing the standard deviation help in understanding the distribution of shoe sizes?
5. What is the effect of outliers on the mean and standard deviation in this case?

**Tip**: Always consider both central tendency and dispersion to get a full understanding of your data's distribution.

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 players? 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 mean, median, mode, range, variance, and standard deviation for the shoe sizes of a basketball team. This problem involves analyzing a data set of shoe sizes (8, 9, 9, 10, 10, 10, 11, 12) and applying measures of central tendency and dispersion. Useful for understanding how statistical measures help in decision-making.

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