Find the following measurements for the data set: 7, 10, 12, 12, 12, 15, 16 mean = median = mode = range = variance = standard deviation =

Answer 1

To find the various measurements for the dataset: 7, 10, 12, 12, 12, 15, 16, we can calculate each statistic step-by-step.

1. Mean: The mean is calculated by summing all the values and dividing by the number of values.

   \[ \text{Mean} = \frac{(7 + 10 + 12 + 12 + 12 + 15 + 16)}{7} = \frac{84}{7} = 12 \]

2. Median: The median is the middle value in an ordered dataset. Since there are 7 numbers (an odd count), the median is the 4th value.

   Ordered data: 7, 10, 12, 12, 12, 15, 16

   \[ \text{Median} = 12 \]

3. Mode: The mode is the value that appears most frequently in the dataset.

   In this data set, 12 appears three times, more than any other number.

   \[ \text{Mode} = 12 \]

4. Range: The range is the difference between the maximum and minimum values.

   \[ \text{Range} = 16 - 7 = 9 \]

5. Variance: Variance measures how much the values in the dataset vary from the mean.

   First, we calculate the deviations from the mean, square them, and then take the average of those squared deviations.

   Deviations:

   • (7 - 12)² = 25
   • (10 - 12)² = 4
   • (12 - 12)² = 0
   • (12 - 12)² = 0
   • (12 - 12)² = 0
   • (15 - 12)² = 9
   • (16 - 12)² = 16

   Sum of squared deviations: \[ 25 + 4 + 0 + 0 + 0 + 9 + 16 = 54 \]

   Variance is the average of these squared deviations: \[ \text{Variance} = \frac{54}{7} \approx 7.71 \]

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

   \[ \text{Standard Deviation} = \sqrt{7.71} \approx 2.78 \]

Summary of the Measurements:

• Mean = 12
• Median = 12
• Mode = 12
• Range = 9
• Variance ≈ 7.71
• Standard Deviation ≈ 2.78