For a set of ungrouped values the following sums are found:
`" "n=15, sumx=480, sumx^(2)=15735`.
Find the standard deviation.

Answer

To find the standard deviation for the given ungrouped data, we can use the formula for standard deviation:

\[ \sigma = \sqrt{\frac{\sum x^2}{n} - \left(\frac{\sum x}{n}\right)^2} \]

Where:
- \( n \) is the number of observations,
- \( \sum x \) is the sum of the observations,
- \( \sum x^2 \) is the sum of the squares of the observations.

Given:
- \( n = 15 \)
- \( \sum x = 480 \)
- \( \sum x^2 = 15735 \)

### Step 1: Calculate \(\frac{\sum x^2}{n}\)

First, we calculate \(\frac{\sum x^2}{n}\):

\[ \frac{\sum x^2}{n} = \frac{15735}{15} \]

Calculating this gives:

\[ \frac{15735}{15} = 1049 \]

### Step 2: Calculate \(\frac{\sum x}{n}\)

Next, we calculate \(\frac{\sum x}{n}\):

\[ \frac{\sum x}{n} = \frac{480}{15} \]

Calculating this gives:

\[ \frac{480}{15} = 32 \]

### Step 3: Calculate \(\left(\frac{\sum x}{n}\right)^2\)

Now, we square the result from Step 2:

\[ \left(\frac{\sum x}{n}\right)^2 = 32^2 = 1024 \]

### Step 4: Substitute values into the standard deviation formula

Now, we substitute the values into the standard deviation formula:

\[ \sigma = \sqrt{1049 - 1024} \]

Calculating this gives:

\[ \sigma = \sqrt{25} \]

### Step 5: Calculate the final value of standard deviation

Finally, we find the square root:

\[ \sigma = 5 \]

Thus, the standard deviation for the given data is \( 5 \).

### Summary of Steps:
1. Calculate \(\frac{\sum x^2}{n}\).
2. Calculate \(\frac{\sum x}{n}\).
3. Square the result from Step 2.
4. Substitute into the formula and simplify.
5. Take the square root to find the standard deviation.