To find the range of data within one standard deviation of the mean, we first need to calculate the mean and standard deviation of the given data set. The data set provided is:
34, 45, 50, 56, 60, 65, 70, 70, 70, 70, 72, 77, 77, 80, 82, 86, 86, 89, 95, 98
Step 1: Calculate the mean.
\[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} \]
Calculating the sum:
\[ 34 + 45 + 50 + 56 + 60 + 65 + 70 + 70 + 70 + 70 + 72 + 77 + 77 + 80 + 82 + 86 + 86 + 89 + 95 + 98 = 1450 \]
Number of data points = 20.
\[ \text{Mean} = \frac{1450}{20} = 72.5 \]
Step 2: Calculate the standard deviation.
First, we calculate the variance. The variance is the average of the squared differences from the mean.
- Calculate the squared differences from the mean:
\[ \text{Squared differences} = (x - \text{Mean})^2 \]
Calculating the squared differences:
- For 34: \( (34 - 72.5)^2 = 1460.25 \)
- For 45: \( (45 - 72.5)^2 = 756.25 \)
- For 50: \( (50 - 72.5)^2 = 506.25 \)
- For 56: \( (56 - 72.5)^2 = 272.25 \)
- For 60: \( (60 - 72.5)^2 = 156.25 \)
- For 65: \( (65 - 72.5)^2 = 56.25 \)
- For 70: \( (70 - 72.5)^2 = 6.25 \)
- For 70: \( (70 - 72.5)^2 = 6.25 \)
- For 70: \( (70 - 72.5)^2 = 6.25 \)
- For 70: \( (70 - 72.5)^2 = 6.25 \)
- For 72: \( (72 - 72.5)^2 = 0.25 \)
- For 77: \( (77 - 72.5)^2 = 20.25 \)
- For 77: \( (77 - 72.5)^2 = 20.25 \)
- For 80: \( (80 - 72.5)^2 = 56.25 \)
- For 82: \( (82 - 72.5)^2 = 90.25 \)
- For 86: \( (86 - 72.5)^2 = 182.25 \)
- For 86: \( (86 - 72.5)^2 = 182.25 \)
- For 89: \( (89 - 72.5)^2 = 272.25 \)
- For 95: \( (95 - 72.5)^2 = 506.25 \)
- For 98: \( (98 - 72.5)^2 = 625.00 \)
- Now sum the squared differences:
\[ 1460.25 + 756.25 + 506.25 + 272.25 + 156.25 + 56.25 + 6.25 + 6.25 + 6.25 + 6.25 + 0.25 + 20.25 + 20.25 + 56.25 + 90.25 + 182.25 + 182.25 + 272.25 + 506.25 + 625.00 = 3942.50 \]
- Divide by the number of data points (N) to get variance:
\[ \text{Variance} = \frac{3942.50}{20} = 197.125 \]
- Calculate the standard deviation by taking the square root of the variance:
\[ \text{Standard Deviation} = \sqrt{197.125} \approx 14.04 \]
Step 3: Find the range of data within one standard deviation of the mean.
The mean is approximately 72.5, and one standard deviation is approximately 14.04.
\[ \text{Lower bound} = \text{Mean} - \text{Standard Deviation} = 72.5 - 14.04 \approx 58.46 \]
\[ \text{Upper bound} = \text{Mean} + \text{Standard Deviation} = 72.5 + 14.04 \approx 86.54 \]
Step 4: Determine which values fall within this range.
Considering values between approximately 58.46 and 86.54, we identify:
- 60, 65, 70, 70, 70, 70, 72, 77, 77, 80, 82, 86, 86
Final Conclusion:
The range of data within one standard deviation of the mean is approximately:
58.46 to 86.54
This range includes the values: 60, 65, 70, 70, 70, 70, 72, 77, 77, 80, 82, 86, 86.