To determine which data items fall within one standard deviation of the mean from the given dataset, we first need to calculate the mean and standard deviation.
Given data set:
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
\[ \text{Mean} = \frac{\text{Sum of all data values}}{\text{Number of data values}} = \frac{34 + 45 + 50 + 56 + 60 + 65 + 70 + 70 + 70 + 70 + 72 + 77 + 77 + 80 + 82 + 86 + 86 + 89 + 95 + 98}{20} \]
Calculating the sum: \[ 34 + 45 + 50 + 56 + 60 + 65 + 70 + 70 + 70 + 70 + 72 + 77 + 77 + 80 + 82 + 86 + 86 + 89 + 95 + 98 = 1316 \]
Now calculate the mean: \[ \text{Mean} = \frac{1316}{20} = 65.8 \]
Step 2: Calculate the Standard Deviation
First, we find the variance, which is the average of the squared differences from the mean. \[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{N} \]
Calculating \((x_i - \text{Mean})^2\): \[ (34 - 65.8)^2 = 1044.64 \ (45 - 65.8)^2 = 435.24 \ (50 - 65.8)^2 = 252.84 \ (56 - 65.8)^2 = 96.04 \ (60 - 65.8)^2 = 33.64 \ (65 - 65.8)^2 = 0.64 \ (70 - 65.8)^2 = 17.64 \ (70 - 65.8)^2 = 17.64 \ (70 - 65.8)^2 = 17.64 \ (70 - 65.8)^2 = 17.64 \ (72 - 65.8)^2 = 37.44 \ (77 - 65.8)^2 = 124.96 \ (77 - 65.8)^2 = 124.96 \ (80 - 65.8)^2 = 201.64 \ (82 - 65.8)^2 = 262.44 \ (86 - 65.8)^2 = 409.64 \ (86 - 65.8)^2 = 409.64 \ (89 - 65.8)^2 = 552.64 \ (95 - 65.8)^2 = 847.84 \ (98 - 65.8)^2 = 1044.64 \]
Now calculate the sum of squared differences: \[ 1044.64 + 435.24 + 252.84 + 96.04 + 33.64 + 0.64 + 17.64 + 17.64 + 17.64 + 17.64 + 37.44 + 124.96 + 124.96 + 201.64 + 262.44 + 409.64 + 409.64 + 552.64 + 847.84 + 1044.64 = 3951.6 \]
Calculating the variance: \[ \text{Variance} = \frac{3951.6}{20} = 197.58 \]
Calculating the standard deviation: \[ \text{Standard deviation} = \sqrt{197.58} \approx 14.04 \]
Step 3: Define One Standard Deviation Range from the Mean
The range within one standard deviation of the mean is: \[ \text{Mean} \pm \text{Standard deviation} = 65.8 \pm 14.04 \]
So, the range is: \[ (65.8 - 14.04, 65.8 + 14.04) = (51.76, 79.84) \]
Step 4: Select values within the range (51.76, 79.84)
Now we filter the original data for items in that range: Values in the range (51.76, 79.84):
- 56
- 60
- 65
- 70
- 70
- 70
- 70
- 72
- 77
- 77
- 80
The items that fall within this range are: 56, 60, 65, 70, 70, 70, 70, 72, 77, 77
Answering the Question
From the answer choices, the correct choice is:
C. 56,60,65,70,72,77,80,82,86 (with a slight discrepancy with 80 and 82 being out of range)
Hence, A is the closest match:
- A. 56,60,65,70,70,70,70,72,77,77 (correct, as it falls under one standard deviation limit).
So the correct answer is A.
3 answers