Chapter B.2, Problem 42E

i. To find Mean, Median & Mode of given data.

To determine

Compare the Central Tendency for the given data

Answer to Problem 42E

• Mean = 32
• Median = 33
• Mode = 32
• Mean, Median and Mode will increase by 6

Explanation of Solution

Given:

Compare the Central Tendency for the given data

• 17,37,32,39,33,34,32
• Add 6 to each value above

Calculation:

(a) Arrange the data in ascending order

17

32

32

33

34

37

39

  $\begin{array}{l}\text{Mean = }\frac{\text{Sum of Observations}}{\text{Number of Observations}}\\ {\text{Mean}}_{(\mathrm{i})}=\frac{17+32+32+33+34+37+39}{7}\\ {\text{Mean}}_{(\mathrm{i})}=\frac{224}{7}=\text{ 32}\end{array}$

  $\begin{array}{l}{\text{Median}}_{(\mathrm{i})}{\text{ = 4}}^{\mathrm{t}\mathrm{h}}\text{ Observation}\\ {\text{Median}}_{(\mathrm{i})}\text{ = 33}\\ {\text{Mode}}_{(\mathrm{i})}\text{ = 32}\end{array}$

(b) Arrange the data in ascending order

23

38

38

39

40

43

45

  $\begin{array}{l}{\text{Mean}}_{(\mathrm{i}\mathrm{i})}=\frac{23+38+38+39+40+43+45}{7}\\ {\text{Mean}}_{(\mathrm{i}\mathrm{i})}=\frac{266}{7}=\text{ 38}\end{array}$

  $\begin{array}{l}{\text{Median}}_{(\mathrm{i}\mathrm{i})}={\text{ 4}}^{\mathrm{t}\mathrm{h}}\text{ observation}\\ {\text{Median}}_{(\mathrm{i}\mathrm{i})}=\text{ 39}\\ {\text{Mode}}_{(\mathrm{i}\mathrm{i})}=\text{38}\end{array}$

Conclusion:

Thus,

Adding 6 to the entire data set also adds 6 to the Mean, Median and Mode.

ii. Effect of constant k on Central Tendency

To determine

Change in central tendency.

Answer to Problem 42E

Central Tendency will increase by constant k.

Explanation of Solution

Given:

Compare the Central Tendency for the given data

• 17,37,32,39,33,34,32
• Add 6 to each value above

Calculation:

Arrange the data in ascending order

23

38

38

39

40

43

45

  $\begin{array}{l}{\text{Mean}}_{(\mathrm{i}\mathrm{i})}=\frac{23+38+38+39+40+43+45}{7}\\ {\text{Mean}}_{(\mathrm{i}\mathrm{i})}=\frac{266}{7}=\text{ 38}\end{array}$

  $\begin{array}{l}{\text{Median}}_{(\mathrm{i}\mathrm{i})}={\text{ 4}}^{\mathrm{t}\mathrm{h}}\text{ observation}\\ {\text{Median}}_{(\mathrm{i}\mathrm{i})}=\text{ 39}\\ {\text{Mode}}_{(\mathrm{i}\mathrm{i})}=\text{38}\end{array}$

Conclusion:

Adding a constant to each data point or subtracting a constant from each data point, the Mean, Median, and Mode will change by the same amount.