Understanding Standard Deviation Calculation for Ages
This solution explains the step-by-step process to calculate the standard deviation for a given set of ages: 25, 30, 35, 40, 45, and 50 years. Standard deviation measures the amount of variation or dispersion of a set of values.
Step 1: Calculate the Mean (Average) of the Ages
First, we find the mean ($\mu$) of the given ages. The mean is the sum of all values divided by the total number of values.
The ages are: 25, 30, 35, 40, 45, 50.
Number of observations ($n$) = 6.
Sum of ages = $25 + 30 + 35 + 40 + 45 + 50 = 225$ years.
The formula for the mean is:
$$ \mu = \frac{\sum x_i}{n} $$
Calculating the mean:
$$ \mu = \frac{225}{6} = 37.5 \text{ years} $$
Step 2: Calculate the Deviations from the Mean
Next, we find the difference between each age and the mean ($\mu$).
| Age ($x_i$) | Deviation ($x_i - \mu$) |
|---|---|
| 25 | $25 - 37.5 = -12.5$ |
| 30 | $30 - 37.5 = -7.5$ |
| 35 | $35 - 37.5 = -2.5$ |
| 40 | $40 - 37.5 = 2.5$ |
| 45 | $45 - 37.5 = 7.5$ |
| 50 | $50 - 37.5 = 12.5$ |
Step 3: Square the Deviations
Now, we square each of the deviations calculated in the previous step.
| Deviation ($x_i - \mu$) | Squared Deviation $(x_i - \mu)^2$ |
|---|---|
| $-12.5$ | $(-12.5)^2 = 156.25$ |
| $-7.5$ | $(-7.5)^2 = 56.25$ |
| $-2.5$ | $(-2.5)^2 = 6.25$ |
| $2.5$ | $(2.5)^2 = 6.25$ |
| $7.5$ | $(7.5)^2 = 56.25$ |
| $12.5$ | $(12.5)^2 = 156.25$ |
Step 4: Calculate the Variance
The variance ($\sigma^2$) is the average of the squared deviations. Since we are given a specific group of 6 people and asked for the standard deviation of *their* ages, we treat this as a population. Therefore, we divide the sum of squared deviations by the number of observations ($n$).
Sum of Squared Deviations = $156.25 + 56.25 + 6.25 + 6.25 + 56.25 + 156.25 = 437.5$
The formula for population variance is:
$$ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} $$
Calculating the variance:
$$ \sigma^2 = \frac{437.5}{6} \approx 72.9167 $$
Step 5: Calculate the Standard Deviation
The standard deviation ($\sigma$) is the square root of the variance.
The formula for population standard deviation is:
$$ \sigma = \sqrt{\sigma^2} $$
Calculating the standard deviation:
$$ \sigma = \sqrt{72.9167} \approx 8.5404 $$
Step 6: Round the Result
We need to round the standard deviation to two decimal places.
$$ \sigma \approx 8.54 $$
Therefore, the standard deviation of the ages is approximately 8.54 years.
Important Questions from Standard Deviation
A cold drink bottling plant fills bottles of 500 ml. capacity with mean of 500 ml. and a standard deviation of 5 ml. Atleast what percentage of bottles would contain cold drink between 490 ml. and 510 ml.?
The mean and standard deviation of 100 terms are 50 and 3, respectively. The sum of squares of the 100 terms is:
If the mean of a random variable X following Poisson distribution is 3, then standard deviation of the distribution is:
If the standard deviation of a population is 100, then based on a sample of size 100, the standard deviation of sample mean is equal to:
What is the median of 8, 5, 7, 9, 11, 6, 10?
