Mean, Median, Mode, Range and Standard Deviation

Q1. Six students obtained the following scores during a mathematics test. 14, 13, 13, 11, 15, and 12. The above score can be used to obtain their mean, median, mode, range, IQR, and standard deviation as follows.

Score (x)

Frequency (f)

fx

d= x-A where A= mean

fd

(fd)2

14

1

14

1

1

1

13

2

26

0

0

0

11

1

11

-2

-2

4

15

1

15

2

2

4

12

1

12

-1

-1

1

∑f= 6

∑fx= 78

∑fd= 10

Measure

dataset

Value

mean

78/6

13

Median

11, 12, 13, 13, 14, 15

13

Mode

11, 12, 13, 13, 14, 15

13

Range

11, 12, 13, 13, 14, 15

11-15

IQR

(11, 12, 13) (13, 14, 15)

14-12

2

Standard deviation (σ)

√ (10/6)

1.414214

An outlier score was added, and the new dataset looked as follows. 14, 13, 13, 5, 15, 12. The new values appeared as follows after determining the measures.

Score (x)

Frequency (f)

fx

d= x-A where A= mean

fd

(fd)2

14

1

14

2

2

4

13

2

26

1

2

4

5

1

5

-7

-7

49

15

1

15

3

3

9

12

1

12

0

0

0

∑f= 6

∑fx= 72

∑= 66

Measure

dataset

Value

Mean

72/6

12

Median

5, 12, 13, 13, 14, 15

13

Mode

5, 12, 13, 13, 14, 15

13

range

5, 12, 13, 13, 14, 15

5-15

IQR

(5, 12, 13) (13, 14, 15)

14-12

2

Standard deviation

√ (66/6)

3.577709

Measure

Dataset 1

Dataset 2

Change

Mean

13

12

Changed (-1)

Median

13

13

No change

Mode

13

13

No change

Range

11-15

5-15

Changed

IQR

2

2

No change

Standard deviation

1.414214

3.577709

Changed (2.163495)

The comparison above shows that the mean, range, and standard deviation were affected by the addition of an outlier. However, median, mode, and IQR remained the same after an outlier was added. Such measures are termed as resistant measures since they remain unchanged after addition of an outlier. In most cases, median and IQR remain the only measures which remain static after an outlier id introduced.

Q 2. Correlation and causation are two statistical variables used interchangeably although their meaning differs. While correlation is used to imply the existence of a close relationship between two or more variables, causation shows the relationship between two variable which is directly connected; occurrence or action of one causes another. For instance, temperatures decrease as one moves up a high mountain. In this case, there exists causation between temperature and altitude. Also, exercising improves the health of an individual. There is causation between health and exercising since there exists a direct connection between the two variables. On the other hand, associating deteriorating mental health with smoking is a correlation since there are other factors such as stress that can cause the same. Causation is used to reaffirm certainty while correlation is used to show a possibility of two variables relating. Another difference between correlation and causation is demonstrated through their techniques of measurement. Correlation is measured using a Pearson’s correlation coefficient while causation is determined through the use of controlled study which can provide statistical information that allows direct determination of data necessary for establishing the existence of causation between two variables. A set of data can show correlation, but fail to prove causation.