Question

Which data set is the most clustered around its mean?
10, 30, 40, 20
12, 14, 36, 38
8, 42, 30, 30
30, 25, 15, 30

Answer 1

To determine which data set is the most clustered around its mean, we can calculate the mean and the variance (or standard deviation) of each data set. A lower variance or standard deviation indicates that the data points are more closely clustered around the mean.

Let's calculate the mean and variance for each data set.

1. **Data Set 1: 10, 30, 40, 20**
- Mean: (10 + 30 + 40 + 20) / 4 = 25
- Variance: [(10-25)² + (30-25)² + (40-25)² + (20-25)²] / (4-1)
= [(225 + 25 + 225 + 25) / 3]
= (500 / 3) ≈ 166.67

2. **Data Set 2: 12, 14, 36, 38**
- Mean: (12 + 14 + 36 + 38) / 4 = 25
- Variance: [(12-25)² + (14-25)² + (36-25)² + (38-25)²] / (4-1)
= [(169 + 121 + 121 + 169) / 3]
= (580 / 3) ≈ 193.33

3. **Data Set 3: 8, 42, 30, 30**
- Mean: (8 + 42 + 30 + 30) / 4 = 27.5
- Variance: [(8-27.5)² + (42-27.5)² + (30-27.5)² + (30-27.5)²] / (4-1)
= [(380.25 + 210.25 + 6.25 + 6.25) / 3]
= (603 / 3) = 201

4. **Data Set 4: 30, 25, 15, 30**
- Mean: (30 + 25 + 15 + 30) / 4 = 25
- Variance: [(30-25)² + (25-25)² + (15-25)² + (30-25)²] / (4-1)
= [(25 + 0 + 100 + 25) / 3]
= (150 / 3) = 50

Now, let's summarize the variances of each data set:

- Data Set 1: Variance ≈ 166.67
- Data Set 2: Variance ≈ 193.33
- Data Set 3: Variance = 201
- Data Set 4: Variance = 50

The data set with the smallest variance is Data Set 4 (30, 25, 15, 30), indicating that it is the most clustered around its mean.