Find the standard deviation (rounded to the nearest unit) for the data indicated.
Test Score Frequency
90 2
80 5
70 9
60 6
50 4
40 1
30 2
0 5
Just pretend that you have:
90,90,80,80,80,80,80, ... ,30,30,0,0,0,0,0
add up the frequencies to find out how many entries you have, that would be 34
mean = sum of all the entries/34
use the method you learned to find the SD of these 34 entries.
To find the standard deviation, you'll need to calculate the variance first. Here are the steps to find the standard deviation:
Step 1: Calculate the mean (average) of the data set. To do this, multiply each test score by its frequency, then sum up the results, and divide by the total frequency.
(90 * 2 + 80 * 5 + 70 * 9 + 60 * 6 + 50 * 4 + 40 * 1 + 30 * 2 + 0 * 5) / (2 + 5 + 9 + 6 + 4 + 1 + 2 + 5) = 58.21 (rounded to two decimal places)
Step 2: Calculate the squared difference between each data point and the mean. To do this, subtract the mean from each test score, then square the result, and multiply by the frequency. Finally, sum up these squared differences.
[(90 - 58.21)^2 * 2 + (80 - 58.21)^2 * 5 + (70 - 58.21)^2 * 9 + (60 - 58.21)^2 * 6 + (50 - 58.21)^2 * 4 + (40 - 58.21)^2 * 1 + (30 - 58.21)^2 * 2 + (0 - 58.21)^2 * 5] = 6982.37 (rounded to two decimal places)
Step 3: Calculate the variance by dividing the sum of squared differences by the total frequency.
6982.37 / (2 + 5 + 9 + 6 + 4 + 1 + 2 + 5) = 410.73 (rounded to two decimal places)
Step 4: Finally, calculate the standard deviation by taking the square root of the variance.
√410.73 ≈ 20.26 (rounded to the nearest unit)
Therefore, the standard deviation (rounded to the nearest unit) for the given data set is approximately 20.
To find the standard deviation of a set of data, you need to follow these steps:
1. Calculate the mean of the data set:
The mean is calculated by adding up all the values and dividing by the total number of values. In this case, the mean can be calculated as follows:
(90 * 2) + (80 * 5) + (70 * 9) + (60 * 6) + (50 * 4) + (40 * 1) + (30 * 2) + (0 * 5) / (2 + 5 + 9 + 6 + 4 + 1 + 2 + 5) = 57.44 (rounded to two decimal places)
2. Calculate the squared difference for each value:
Subtract the mean from each value, then square the result. For example, for the first value (90), the squared difference would be (90-57.44)^2 = 1007.36. Compute the squared difference for each value in the data set.
3. Calculate the sum of the squared differences:
Add up all the squared differences from step 2.
4. Calculate the variance:
Divide the sum of the squared differences from step 3 by the total number of values. In this case, the variance can be calculated as follows:
Sum of squared differences / Total number of values = Variance
Variance = (1007.36 + 166.24 + 4.84 + 67.24 + 1.44 + 2420.24 + 529.04 + 6.76) / (2 + 5 + 9 + 6 + 4 + 1 + 2 + 5) = 305.50 (rounded to two decimal places)
5. Calculate the standard deviation:
The standard deviation is the square root of the variance. In this case, the standard deviation can be calculated as follows:
Square root of the variance = √305.50 = 17.48 (rounded to two decimal places)
Therefore, the standard deviation (rounded to the nearest unit) for the given data set is approximately 17.