Range and Deviation?

Question

Find the range for each group of data items: 
a. 1, 2, 3, 4, 5 
b. 3, 3, 4, 4, 5, 5 

Find the standard deviation for each group of data items. Round answers to two decimal places. 
c. 1, 2, 3, 4, 5 
d1, 1, 1, 4, 7, 7, 7

Answer 1

Note :

• Statistical range is a difference between the maximum value and minimum value in the set of numbers.
• There are two steps to find the range of set of numbers.
• Arrange the numbers in ascending or descending order by size or subtract minimum value by maximum value.
• For example, if the highest value of set of numbers is 100 and lowest value of set of numbers is 10, then the range is 100 -10 = 90.

a). 1, 2, 3, 4, 5.

The above data items are in ascending order.

The highest value of set of items is 5, and

The lowest value of set of items is 1.

The range is 5 - 1 = 4.

b). 3, 3, 4, 4, 5, 5.

The above data items are in ascending order.

The highest value of set of items is 5, and

The lowest value of set of items is 3.

The range is 5 - 3 = 2.

Answer 2

Standard deviation :

1). The first step in calculating a standard deviation is to find the mean in a set of data.

2). The next step is to get the deviations in these numbers. Do this by subtracting the mean from each of the numbers in the set of data.

3). Now, square the deviations calculated in Step 2. To "square" a number simply means to multiply the number by itself. Remember that when you square a negative number, the result is a positive number.

4). Next, add all of the squares in Step 3 together.

5). Now divide the sum in Step 4 by the total number of numbers, less one.

6). Take the square root of the result in Step 5. The square root is the number, when multiplied by itself, equals the original number.Thus, the standard deviation of this set of numbers.

(C). 1, 2, 3, 4, 5.

Step 1 :

The mean of the set of numbers = (1 + 2 +3 + 4 + 5)/5 = 15/5 = 3.

Step 2 :

Subtracting the mean from each of the numbers in the set of data.

1 - 3 = - 2

2 - 3 = - 1

3 - 3 = 0

4 - 3 = 1

5 - 3 = 2.

Step 3 :

Now, square the deviations calculated in Step 2.

(- 2)² = 4

(- 1)² = 1

(0)² = 0

(1)² = 1

(2)² = 4.

Step 4 :

Next, add all of the squares in Step 3 together.

4 + 1 + 0 + 1 + 4 = 10.

Step 5 :

Now divide the sum in Step 4 by the total number of numbers, less one.

10/4 = 5/2 = 2.5 .

Step 6 :

Take the square root of the result in Step 5

√2.5 = 1.5811.

Thus, the standard deviation of this set of numbers is 1.5811.

Answer 3

(d). 1, 1, 1, 4, 7, 7, 7.

Step 1 :

The mean of the set of numbers = (1 + 1 + 1 + 4 + 7 + 7 + 7)/7 = 28/7 = 4.

Step 2 :

Subtracting the mean from each of the numbers in the set of data.

1 - 4 = - 3

1 - 4 = - 3

1 - 4 = - 3

4 - 4 = 0

7 - 4 = 3

7 - 4 = 3

7 - 4 = 3.

Step 3 :

Now, square the deviations calculated in Step 2.

(- 3)² = 9

(- 3)² = 9

(- 3)² = 9

(0)² = 0

(3)² = 9

(3)² = 9

(3)² = 9.

Step 4 :

Next, add all of the squares in Step 3 together.

9 + 9 + 9 + 0 + 9 + 9 + 9 = 54.

Step 5 :

Now divide the sum in Step 4 by the total number of numbers, less one.

54/6 = 9 .

Step 6 :

Take the square root of the result in Step 5

√9 = 3.

Thus, the standard deviation of this set of numbers is 3.