Calculate the mean, median, mode, range, variance and standard deviation of the distribution

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

1.  The data below shows the number of prisoners who escaped from 10 prisons over a 5 year period. Hand Calculate the mean, median, mode, range,  variance and standard deviation of the distribution

(SHOW YOUR WORK / CALCULATIONS)

Mode = 5,7

Median=5

Mean= 4.8

Range=7

Variance=39.47

Standard deviation = 6.28 

X         X        

2          1          1

4          2          2

5          3          3

3          4          4

7          5          5

7          5          6

5          6          7

6          7          8

1          7          9

8          8          10

 

A.      MODE:

The mode is defined AS THE MOST FREQUENTLY OCCURRING SCORE in a group of scores. 

Mode = 5,7

 

B.      MEDIAN:

( N + 1) / 2

Step 1:  Arrange the numbers in ascending or descending order.

 

Step 2: Find the median position

(10+1)/2 = (11)/2 = 5.5

 

Step  3: So the median position is between 5th and 6th number

The number between the 5th and 6th number is 5 and 5

 Therefore,

 

Median=5

C.      MEAN:

 

WE ARE ONLY DEALING WITH SAMPLE MEAN IN CHAPTER 4 AND 5.

A STATISTIC is a numerical descriptive measure of a sample.

Mean = ∑X / n

Where ∑X is the sum of all values in the data set and

n is the sample size.

∑X=2+4+5+3+7+7+5+6+1+8= 48

N=10

Mean=48/10

Mean= 4.8

  

D.      RANGE

Range simply is defined as the difference between the highest and the lowest value.

Range=8-1=7

Range=7

 E.       Variance

 

 Step 1:  Calculate mean.

 

(Mean = ∑X / n Where ∑X is the sum of all values in the data set and

n is the sample size.)

Mean=4.8

Step 2: Form a column to calculate (X - ). (X - ) is referred to as mean deviation

X         (X -)                                  

1          (1-4.8=--3.8)

2          (2-4.8=--2.8)

3          (3-4.8=-1.8)

4          (4-4.8-=-0.8)

5          (5-4.8=0.2)

5          (5-4.8=0.2)

6          (6-4.8=1.2)

7          (7-4.8=2.2)

7          (7-4.8=2.2)

8          (8-4.8=3.2)

Step 4: Find the total or ∑(X - )2

X         (X - )                                   (X - )2

1          (1-4.8=--3.8)                -3.82=11.24

2          (2-4.8=--2.8)                -2.82=7.84

3          (3-4.8=-1.8)                 -1.82=3.24

4          (4-4.8-=-0.8)                -0.82=0.64

5          (5-4.8=0.2)                  0.22= 0.04

5          (5-4.8=0.2)                  0.22= 0.04

6          (6-4.8=1.2)                  1.22= 1.44

7          (7-4.8=2.2)                  2.22=4.84

7          (7-4.8=2.2)                  2.22=4.84

8          (8-4.8=3.2)                  3.22=10.24

∑ =44.4

 Step 5: calculate s2 = ∑(X - )2 / (n-1).  We are calculating the sample variance so divide the total squared deviation by (n-1).

 

NOTE: The symbol for population variance is σ2 and for the sample variance it is s2. 

 

s2 = ∑(X - )2 / (n-1)

s2 = 44.4/(10-1)

s2 =44.4/9

s2 =39.466=39.47

The calculated variance for this distribution is 39.47

Variance=39.47

 

F.       Standard Deviation

The calculation of standard deviation is very simple. 

Simply take the square root of the variance.

 Standard deviation = s = √39.47 = 6.282=6.28

 

Standard deviation = 6.28 

2.       Hand calculate mean, median, mode, range, variance and standard deviation for the following distribution:  X represents how many brothers and sisters do you have and f represents frequency (total  N =338)

(SHOW YOUR WORK / CALCULATIONS) 

X             f              fx            cf

0              39           0              39

1              137         137         176

2              79           158         255

3              39           117         294

4              17           68           311

5              13           65           324

6              6              36           330

7              4              38           334

8              1              8              335        

9              1              9              336

12           1              12           337

15           1              15           338

                                ∑663      ∑338

(N =338)

A.      Mean =1.96

B.      Median Position =168.5

C.      Median =1

D.      Mode =1

E.       Range =15

F.       Variance=3.16

G.     Standard deviation=1.78

 A.      Mean:

 

Mean = ∑X / n

Where ∑X is the sum of all values in the data set and

n is the sample size.

∑X=663

N=338

Mean = ∑X / n

Mean=663/338

Mean=1.961

Mean=1.96 

B.      Median Position & C. Median

Cumulative frequency is denoted by cf. 

The data is arranged in descending order. 

Create a cumulative frequency column.   

X   f              Cumulative Frequency                                                 

15               1              39+137+79+39+17+13+6+4+1+1+1+1=338

12               1              39+137+79+39+17+13+6+4+1+1+1=337

9  1              39+137+79+39+17+13+6+4+1+1=336

8  1              39+137+79+39+17+13+6+4+1=335

7  4              39+137+79+39+17+13+6+4=334

6  6              39+137+79+39+17+13+6=330

5  13           39+137+79+39+17+13=324

4  17           39+137+79+39+17=311

3  39           39+137+79+39=294

2  79           39+137+79=255

1  137         39+137=176

0  39           39

Once you get the cumulative frequency you have to find the mid point or median position. 

The median position can be find out by using the formula (N + 1) / 2

(N =338)

Therefore, (N + 1) / 2= (338+1)/2=337/2=168.5

Check where the 168.5 number lies in cumulative frequency. 

The Median Position is 168.5

The 168.5 score is 1. 

SO median is 1. 

 D.      Mode

 

The mode is defined AS THE MOST FREQUENTLY OCCURRING SCORE in a group of scores. 

The most frequently occurring score is 1 with a frequency of 137.

Mode=1

 

Attachments:

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