Mean Median and Mode

Find the mean annual salary for a team put together by a company, where their respective annual salaries are as follows; £22,000, £45,000, £36,800, £70,000, £55,500, and £48,700.

Solution

We sum up the data values and divide them by the number of data values we have, as the formula says.

\[\begin{align}\mu&=\dfrac{\sum x_i}{N}=\\&=\dfrac{£\,22,000+£\,45,000+£36,800+£\,70,000+£\,55,500+£\,48,700}{6}=\\&=\dfrac{£\,278,000}{6}=\\&=£\,46,333.33\end{align}\]

By this calculation, it means that the mean salary amongst the team is £46,333.

Find the mean of the data of salaries of a team of employees put together by a company including their supervisor as £22,000, £45,000, £36,800, £40,000, £70,000, £55,500, and £48,700, find the median.

Solution

We arrange our data values from lowest to the highest.

£22,000, £36,800, £40,000, £45,000, £48,700, £55,500, and £70,000.

We notice that the number of the data values is 7, which is an odd number, so the median is the middle between the lowest half (constituting of £22,000, £36,800, £40,000), and the highest half of the data set (constituting of £48,700, £55,500, and £70,000).

Thus, the middle value here is £45,000 , hence we deduce that

\[\text{Median}=£\,45,000\]

The data set of the team put together by the company excluding their supervisor is as follows, £22,000, £45,000, £36,800, £40,000, £55,500, and £48,700, find the median.

Solution

We arrange these values from the lowest to the highest.

£22,000, £36,800, £40,000, £45,000, £48,700, £55,500.

We notice that the number of the data values is 6, which is an even number, so we have two numbers as our middle data point. Yet, to find the median, we find the average of those two numbers, £40,000 and £45,000.

\[\text{Average}=\dfrac{£\,40,000+£\,45,000}{2}=\dfrac{£\,85,000}{2}=£\,42,500\]

Find the mode for the given data set, 45, 63, 1, 22, 63, 26, 13, 91, 19, 47.

Solution

We rearrange the data set from the lowest to the highest values.

1, 13, 19, 22, 26, 45, 47, 63, 63, 91

We count the occurrence of each data value and we see that all data values occur only once, while the data value 63 occurs twice. Thus the mode of the data set is

\[\text{Mode}=63\]

Suppose Mike wants to buy a property in London so he goes out to find out the prices of what exactly he might like. The data he gets on the pricing of all the properties he enquired about are as follows; £422,000, £250,000, £340,000, £510,000, and £180,000.

Find

1. Mean
2. Median
3. Mode

Solution

1. To find the mean, we use the mean formula. We first find the sum of all the data values and divide it by the number of data values.

\[\mu=\dfrac{\sum x_1}{N}=\dfrac{£\,422,000+£\,250,000+£\,340,000+£\,510,000+£\,180,000}{5}\]

\[\mu=\dfrac{£\,1,702,00}{5}=£\,340,400\]

The mean price is £340,400

2. To find the median, we will need to arrange the data values in ascending order,

£180,000, £250,000, £340,000, £422,000, £510,000 .

The number of the data values is 5, which is odd, so we notice that the third data value is the middle between the lowest half and the highest half. So, we can now easily identify what the middle point value is

\[\text{Median}=£\,340,000\}

3. The mode is the most occurred data value. To find it, we will first rearrange the data values in ascending order.

£180,000, £250,000, £340,000, £422,000, £510,000

We notice that there is no most occurred data value. Thus, the data set has no mode.

The heights of students in grade 11 were collected and the data is given as

173cm, 151cm, 160cm, 151cm, 166cm, 149cm.

Find

1. Mean
2. Median
3. Mode

Solution

1. To find the mean, we will use the mean formula, in which we add all the data values and divide the sum by the number of data values.

\[\begin{align}\mu&=\dfrac{\sum x_i}{N}=\dfrac{173\,\mathrm{cm}+151\,\mathrm{cm}+160\,\mathrm{cm}+151\,\mathrm{cm}+166\,\mathrm{cm}+149\,\mathrm{cm}}{6}=\\\\&=\dfrac{950\,\mathrm{cm}}{6}=158.33\,\mathrm{cm}\end{align}\]

The mean height is \(158.33\,\mathrm{cm}\).

2. The median is the middle point value of the data set. To find it, we will rearrange the data values in ascending order first, to get

149 cm, 151 cm, 151 cm, 160 cm, 166 cm, 173 cm

We notice that the number of the data values is 6, which is an even number, and hence we have two values in the middle. They are 151 cm and 160 cm. We will find the average of these values by adding them and dividing them by 2.

\[\dfrac{151+160}{2}=\dfrac{311}{2}=155.5\]

Thus, the median is

\[\text{Median}=155.5\,\mathrm{cm}\]

3. The mode is the most occurring value in the data set. We can rearrange the data values in ascending order to get,

149 cm, 151 cm, 151 cm, 160 cm, 166 cm, 173 cm.

We can identify that 151cm is the most commonly occurring value, thus

\[\text{Mode}=151\,\mathrm{cm}\]