Find the standard deviation (rounded to the nearest unit) for the data indicated.
Test Score Frequency
90 1
80 4
70 7
60 6
50 3
The standard deviation
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Calculate, correct to the nearest unit
To find the standard deviation, you need to follow these steps:
Step 1: Calculate the mean (average) of the data.
To do this, you will need to multiply each test score by its frequency, then add up the results, and finally divide by the total number of data points.
Mean = (90*1 + 80*4 + 70*7 + 60*6 + 50*3) / (1+4+7+6+3)
Step 2: Calculate the deviation of each data point from the mean.
To do this, subtract the mean from each test score.
Deviation = Test Score - Mean
Step 3: Square each deviation.
To remove the negative signs and emphasize larger deviations, square each deviation.
Squared Deviation = Deviation^2
Step 4: Calculate the variance.
To find the variance, add up all the squared deviations and divide by the total number of data points.
Variance = (Squared Deviation1 + Squared Deviation2 + ... + Squared Deviationn) / n
Step 5: Calculate the standard deviation.
To find the standard deviation, take the square root of the variance.
Standard Deviation = √Variance
Now let's calculate the standard deviation for the given data:
Step 1:
Mean = (90*1 + 80*4 + 70*7 + 60*6 + 50*3) / (1+4+7+6+3) = 65.75 (rounded to two decimal places)
Step 2:
Deviation = [90-65.75, 80-65.75, 70-65.75, 60-65.75, 50-65.75]
Deviation = [24.25, 14.25, 4.25, -5.75, -15.75]
Step 3:
Squared Deviation = [24.25^2, 14.25^2, 4.25^2, (-5.75)^2, (-15.75)^2]
Squared Deviation = [588.06, 203.06, 18.06, 33.06, 248.06]
Step 4:
Variance = (588.06 + 203.06 + 18.06 + 33.06 + 248.06) / 21 ≈ 29.43 (rounded to two decimal places)
Step 5:
Standard Deviation = √(29.43) ≈ 5.43 (rounded to the nearest unit)
Therefore, the standard deviation for the given data is approximately 5.