Connecting the Dots… | IT

Question 15

Context: Poovizhi's family heights are 118 cm, 165 cm, 170 cm, 173 cm, and 175 cm.

Q. Find the mean and median in Poovizhi's data without the outlier value 118. What change do you notice?

Solution

We will find the mean and median of the heights, first with the outlier, then without it.

Step 1 — List the heights

Poovizhi's family heights are given. We list them in increasing order.

$118 \text{ cm, } 165 \text{ cm, } 170 \text{ cm, } 173 \text{ cm, } 175 \text{ cm}$

The value 118 cm is much smaller than others. This value is an outlier. We will remove it for the second part. The heights without the outlier are:

$165 \text{ cm, } 170 \text{ cm, } 173 \text{ cm, } 175 \text{ cm}$

Step 2 — Calculate original mean and median

First, let us find the mean of all five heights. The mean is the sum of all values divided by the number of values. The sum of the heights is:

$118 + 165 + 170 + 173 + 175$

$= 801$

There are 5 heights. So, the original mean is:

$\frac{801}{5}$

$= 160.2$

$\boxed{\text{Original Mean} = 160.2 \text{ cm}}$

Now, let us find the median of all five heights. The median is the middle value when data is ordered. Our data is already ordered: 118, 165, 170, 173, 175. There are 5 values. The middle value is the 3rd value.

$\boxed{\text{Original Median} = 170 \text{ cm}}$

Step 3 — Calculate new mean (without outlier)

Now, we remove the outlier 118 cm. The new heights are 165 cm, 170 cm, 173 cm, 175 cm. Let us find the sum of these new heights.

$165 + 170 + 173 + 175$

$= 683$

There are 4 new heights. So, the new mean is:

$\frac{683}{4}$

$= 170.75$

$\boxed{\text{New Mean} = 170.75 \text{ cm}}$

Step 4 — Calculate new median (without outlier)

Let us find the median of the new heights. The new heights are 165, 170, 173, 175. There are 4 values. Since there is an even number of values, we take the average of the two middle values. The two middle values are the 2nd and 3rd values. These are 170 and 173. The average of these two values is:

$\frac{170 + 173}{2}$

$\frac{343}{2}$

$= 171.5$

$\boxed{\text{New Median} = 171.5 \text{ cm}}$

Step 5 — Notice the change

We compare the original values with the new values. The original mean was 160.2 cm. The new mean is 170.75 cm. The mean increased by $170.75 - 160.2 = \textbf{10.55 cm}$ .

The original median was 170 cm. The new median is 171.5 cm. The median increased by $171.5 - 170 = \textbf{1.5 cm}$ .

The mean changed much more than the median. The outlier pulled the mean down significantly.

Answer

(i) The mean without the outlier is 170.75 cm. (ii) The median without the outlier is 171.5 cm. (iii) The mean changed by 10.55 cm, while the median changed by 1.5 cm. The mean was affected much more by removing the outlier.

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Context: The heights of the family members of Yaangba and Poovizhi are as follows:

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Q13

Context: The heights of the family members of Yaangba and Poovizhi are: Yaangba's family: 169 cm, 173 cm, 155 cm, 165 cm, 160 cm, 164 cm. Poovizhi's family: 170 cm, 173 cm, 165 cm, 118 cm, 175 cm.

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Q14

Context: The heights of the family members of Yaangba and Poovizhi are: Yaangba's family: 169 cm, 173 cm, 155 cm, 165 cm, 160 cm, 164 cm. Poovizhi's family: 170 cm, 173 cm, 165 cm, 118 cm, 175 cm.

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Q15

Context: Poovizhi's family heights are 118 cm, 165 cm, 170 cm, 173 cm, and 175 cm.

Q. Find the mean and median in Poovizhi's data without the outlier value 118. What change do you notice?

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Q39

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