A random sample of 40 smokers is classified in the following table:
Ages Frequency
[10-20) 4
[20-30) 6
[30-40) 12
[40-50) 10
[50-60) 8
Total 40
The standard deviation is?
13.6491
14.0264
12.4448
11.8707
To calculate the standard deviation, we first need to find the mean.
Mean = Σ (midpoint of each interval * frequency) / Total frequency
Mean = [(15 * 4) + (25 * 6) + (35 * 12) + (45 * 10) + (55 * 8)] / 40
Mean = (60 + 150 + 420 + 450 + 440) / 40
Mean = 1520 / 40
Mean = 38
Next, we calculate the squared difference between each midpoint and the mean, multiply by the frequency, sum them all up and divide by total frequency -1 to find the variance.
Variance = Σ [ (midpoint - mean)^2 * frequency ] / (Total frequency - 1)
Variance = [ (15 - 38)^2 * 4 + (25 - 38)^2 * 6 + (35 - 38)^2 * 12 + (45 - 38)^2 * 10 + (55 - 38)^2 * 8 ] / 39
Variance = [ (-23)^2 * 4 + (-13)^2 * 6 + (-3)^2 * 12 + (7)^2 * 10 + (17)^2 * 8 ] / 39
Variance = [ 529 * 4 + 169 * 6 + 9 * 12 + 49 * 10 + 289 * 8 ] / 39
Variance = [ 2116 + 1014 + 108 + 490 + 2312 ] / 39
Variance = 6040 / 39
Variance = 154.87
Finally, we calculate the standard deviation by taking the square root of the variance.
Standard deviation = √Variance
Standard deviation ≈ √154.87
Standard deviation ≈ 12.4448
Therefore, the standard deviation is approximately 12.4448.