Final Answer:
For the data set {3, 9, 10, 5, 8, 5, 9, 11, 10, 3, 10}:
a. The mean of the data is 7.55.
b. The median of the data is 9.
c. The mode of the data is 10.
d. The range of the data is 8.
e. The standard deviation of the data is 2.98.
Step-by-step explanation:
a. Mean:
To find the mean, add all the numbers and divide by the total number of numbers:
Mean = (3 + 9 + 10 + 5 + 8 + 5 + 9 + 11 + 10 + 3 + 10) / 11
= 78 / 11
= 7.55
b. Median:
First, order the data from least to greatest:
3, 3, 5, 5, 8, 9, 9, 10, 10, 10, 11
Since we have an odd number of data points, the median is the middle value:
Median = 9
c. Mode:
The mode is the most frequent value in the data set. In this case, 10 appears three times, making it the mode:
Mode = 10
d. Range:
The range is the difference between the highest and lowest values:
Range = 11 (highest) - 3 (lowest)
= 8
e. Standard deviation:
The standard deviation is a measure of how spread out the data is from the mean. Here, we'll calculate the sample standard deviation:
1. Calculate the squared deviations from the mean for each data point:
(3 - 7.55)^2 = 19.36
(9 - 7.55)^2 = 2.25
(10 - 7.55)^2 = 5.76
(5 - 7.55)^2 = 6.25
(8 - 7.55)^2 = 0.25
(5 - 7.55)^2 = 6.25
(9 - 7.55)^2 = 2.25
(11 - 7.55)^2 = 12.56
(10 - 7.55)^2 = 5.76
(3 - 7.55)^2 = 19.36
(10 - 7.55)^2 = 5.76
2. Sum the squared deviations from the mean:
19.36 + 2.25 + 5.76 + 6.25 + 0.25 + 6.25 + 2.25 + 12.56 + 5.76 + 19.36 + 5.76 = 85.8
3. Divide the sum of squared deviations from the mean by the number of data points minus 1:
85.8 / (11 - 1) = 8.58
4. Take the square root of the result:
√8.58 ≈ 2.98
Therefore, the standard deviation of the data is approximately 2.98.