Practice Sheet Mean, Median, Mode, Variance and Standard ...

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Practice Sheet Mean, Median, Mode, Variance and Standard

Deviation

Population ()

1, 2, 3, , -2, -1,

Sample of the population ()

1, 2, 3, , -2, -1,

Population Mean () Sample Mean ()

=

=

=1

=

=

=1

Median Arrange data in ascending order. ( or can be used depending on the data.)

1, 2, 3, , -2, -1,

1 2 3 -2 -1 If there are an odd number of data values, the median is the middle number (/2+0.5). If there is an even number of data values, the median is the average of the two middle numbers (/2+2/2+1).

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Mode Mode is the values that occur the most often. If there are more than one mode, then the data is multi modal. If there are no common values, then the data has no mode.

Population Variance ()

2

=

=1( -

)2

Sample Variance ()

2

=

=1( - - 1

)2

Population Standard Deviation ()

=

=

=1(

-

)2

Sample Standard Deviation ()

=

=

=1( - - 1

)2

Example 1

9, 4, 8, 12, 5, 1, 8, 18, 9, 13, 18, 1, 5, 10, 15, 8, 15, 8, 8, 5

Assuming the data above is the total population. What is the a) mean, b) median, c) mode, d) variance and e) standard deviation? (All answers are to be rounded to 4 decimal places)

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Population Mean

Step 1: Input the data and information into the mean equation and calculate. = 20.

=

= 1

9 + 4 + 8 + 12 + 5 + 1 + 8 + 18 + 9 + 13 + 18 + 1 + 5 + 10 + 15 + 8 + 15 + 8 + 8 + 5

=

20

180 = 20

=9

The mean is 9.

Median

Step 1: Arrange the data in ascending order.

1, 1, 4, 5, 5, 5, 8, 8, 8, 8,8, 9, 9, 10, 12, 13, 15, 15, 18, 18

Step 2: If there are an odd number of data values, the median is the middle number (/2+0.5). If

there is an even number of data values, the median is the average of the two middle numbers (/2+2/2+1). In this example = 20 which is even so the average of the two middle numbers must be found.

Step 3: Find the two middle numbers /2 and /2+1, /2 = 10 and /2 + 1 = 11.

1, 1, 4, 5, 5, 5, 8, 8, 8, 8, 8, 9, 9, 10, 12, 13, 15, 15, 18, 18

1 10

11 20

/2 = 10 =8

and

/2+1 = 11 =8

Step 4: Find the average between the two numbers.

/2

+ /2+1 2

=

10

+ 2

11

8+8 =2

16 =2

=8

The median is 8.

Mode

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1, 1, 4, 5, 5, 5, 8, 8, 8, 8,8, 9, 9, 10, 12, 13, 15, 15, 18, 18

Step 1: Find the most common number. If there is not common value, then there is no mode. If there are values that are tied for the most common value, then there is more than one mode and the data is known to be multi modal. Find the most common value in this data set is 8.

The mode is 8.

Population Variance

Step 1: Input the data and information into the equation population variance equation and calculate. = 2.

2

=

= 1( -

)2

(1 - 9)2 + (1 - 9)2 + (4 - 9)2 + (5 - 9)2 + (5 - 9)2 + (5 - 9)2 + (8 - 9)2 + (8 - 9)2

+(8 - 9)2 + (8 - 9)2 + (8 - 9)2 + (9 - 9)2 + (9 - 9)2 + (10 - 9)2 + (12 - 9)2

=

+(13 - 9)2 + (15 - 9)2 + (15 - 9)2 + (18 - 9)2 + (18 - 9)2 20

(-8)2 + (-8)2 + (-5)2 + (-4)2 + (-4)2 + (-4)2 + (-1)2 + (-1)2

+(-1)2 + (-1)2 + (-1)2 + (0)2 + (0)2 + (1)2 + (3)2

=

+(4)2 + (6)2 + (6)2 + (9)2 + (9)2 20

64 + 64 + 25 + 16 + 16 + 16 + 1 + 1

+1 + 1 + 1 + 0 + 0 + 1 + 9

=

+16 + 36 + 36 + 81 + 81 20

466

= 20

= 23.3

The population variation is approximately 23.3.

Population Standard Deviation

Step 1: Input the data and information into the equation population standard deviation equation. Since the variance was found earlier, 23.3, it does not need to be recalculated.

=

=

=1(

-

)2

= 23.3 4.827007354458868027217

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4.8270 Step 2: Ensure the value is rounded to there required decimal place. In this example it was 4 decimal places.

The population standard deviation is approximately 39.3043.

Example 2

19, 0, 14, 10, 17, 13, 4, 5, 13, 18, 18, 5, 17, 13, 10, 14, 12, 3, 2, 1 Assuming the data above is a sample of the population. What is the a) mean, b) median, c) mode, d) variance and e) standard deviation? (All answers are to be rounded to 4 decimal places)

Sample Mean

Step 1: Input the data and information into the mean equation and calculate. = 20.

=

=1

9 + 4 + 8 + 12 + 5 + 1 + 8 + 18 + 9 + 13 + 18 + 1 + 5 + 10 + 15 + 8 + 15 + 8 + 8 + 5

=

20

180 = 20

=9

The mean is 9.

Median

Step 1: Arrange the data in ascending order.

0, 1, 2, 3, 4, 5, 5, 10, 10, 12, 13, 13, 13, 14, 14, 17, 17, 18, 18, 19

Step 2: If there are an odd number of data values, the median is the middle number (/2+0.5). If

there is an even number of data values, the median is the average of the two middle numbers (/2+2/2+1). In this example = 20 which is even so the average of the two middle numbers must be found.

Step 3: Find the two middle numbers /2 and /2+1, /2 = 10 and /2 + 1 = 11.

0, 1, 2, 3, 4, 5, 5, 10, 10, 12, 13, 13, 13, 14, 14, 17, 17, 18, 18, 19

1 10

11 20

/2 = 10

= 12

and

/2+1 = 11

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= 13

/2-0.5

+ 2

/2+0.5

=

12

+ 2

13

15 =2

= 7.5

The median is 7.5.

Mode

0, 1, 2, 3, 4, 5, 5, 10, 10, 12, 13, 13, 13, 14, 14, 17, 17, 18, 18, 19

Step 1: Find the most common number. If there is not common value, then there is no mode. If there are values that are tied for the most common value, then there is more than one mode and the data is known to be multi modal. Find the most common value in this data set is 13.

The mode is 13.

Sample Variance

Step 1: Input the data and information into the equation population variance equation and calculate.

2

=

=1( - - 1

)2

(0 - 10.4)2 + (1 - 10.4)2 + (2 - 10.4)2 + (3 - 10.4)2 + (4 - 10.4)2 + (5 - 10.4)2 + (5 - 10.4)2

+(10 - 10.4)2 + (10 - 10.4)2 + (12 - 10.4)2 + (13 - 10.4)2 + (13 - 10.4)2 + (13 - 10.4)2

=

+(14

-

10.4)2

+

(14

-

10.4)2

+

(17

-

10.4)2

+

(17 - 10.4)2 20 - 1

+

(18

-

10.4)2

+

(18

-

10.4)2

+

(19

-

10.4)2

(-10.4)2 + (-9.4)2 + (-8.4)2 + (-7.4)2 + (-6.4)2 + (-5.4)2 + (-5.4)2

+(-0.4)2 + (-0.4)2 + (1.6)2 + (2.6)2 + (2.6)2 + (2.6)2

=

+(3.6)2 + (3.6)2 + (6.6)2 + (6.6)2 + (7.6)2 + (7.6)2 + (8.6)2 19

108.16 + 88.36 + 70.56 + 54.76 + 40.96 + 29.16 + 29.16

+0.16 + 0.16 + 2.56 + 6.76 + 6.76 + 6.76

=

+12.96

+

12.96

+

43.56

+

43.56 19

+

57.76

+

57.76

+

73.96

746.8

= 19

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39.30526316 39.3053 Step 1: Ensure that the answer is rounded to the required decimal places. In this example it is 4 decimal places.

The sample variance is approximately 39.3053.

Sample Standard Deviation

=

=

=1( - - 1

)2

=

746.8 19

6.269390972 6.2694

The standard deviation is approximately 6.2694.

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Questions

(All answers are to be rounded to 4 decimal places)

20, 12, 17, 13, 18

1)

Assuming the data above is the total population. What is the a) mean, b) median,

c) mode, d) variance and e) standard deviation?

7, 7, 10, 5, 17, 16, 8, 1, 9, 15, 3, 3, 16, 16

2)

Assuming the data above is the total population. What is the a) mean, b) median,

c) mode, d) variance and e) standard deviation?

1, 14, 20, 13, 17, 4, 20

3)

Assuming the data above is the total population. What is the a) mean, b) median,

c) mode, d) variance and e) standard deviation?

12, 16, 0, 17, 6, 12, 15, 10, 6, 14

4)

Assuming the data above is the total population. What is the a) mean, b) median,

c) mode, d) variance and e) standard deviation?

18, 8, 9, 8, 14, 4, 11, 20, 19, 7, 6, 9, 17, 8, 5, 20, 6, 20

5)

Assuming the data above is the total population. What is the a) mean, b) median,

c) mode, d) variance and e) standard deviation?

6, 9, 6, 10, 7, 20, 3, 9, 11, 10, 3, 9, 15, 10, 1, 1, 13, 9, 12, 13

6)

Assuming the data above is the total population. What is the a) mean, b) median,

c) mode, d) variance and e) standard deviation?

20, 19, 1, 3, 12, 10, 2, 8, 2, 1, 10, 20, 2

7)

Assuming the data above is the total population. What is the a) mean, b) median,

c) mode, d) variance and e) standard deviation?

18, 11, 16, 2, 4, 15, 7, 3, 4, 1, 14, 16, 14, 10, 9, 10, 9, 12, 4, 19

8)

Assuming the data above is the total population. What is the a) mean, b) median,

c) mode, d) variance and e) standard deviation?

20, 7, 0, 12, 18, 1, 14, 16, 2, 15, 4, 16, 14, 3, 9, 7, 12, 19, 14, 15

9)

Assuming the data above is the total population. What is the a) mean, b) median,

c) mode, d) variance and e) standard deviation?

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