Mode

Find the mode of the data set \(\{2; 2; 3; 4; 4; 4; 6; 6; 7; 8; 8; 10; 10\}\).

Count the number of times that each value appears in the data set

Value	Count
\(\text{2}\)	\(\text{2}\)
\(\text{3}\)	\(\text{1}\)
\(\text{4}\)	\(\text{3}\)
\(\text{6}\)	\(\text{2}\)
\(\text{7}\)	\(\text{1}\)
\(\text{8}\)	\(\text{2}\)
\(\text{10}\)	\(\text{2}\)

Find the value that appears most often

From the table above we can see that \(\text{4}\) is the only value that appears \(\text{3}\) times. All the other values appear less than 3 times. Therefore the mode of the data set is \(\text{4}\).

There are regulations in South Africa related to bread production to protect consumers. By law, if a loaf of bread is not labelled, it must weigh \(\text{800}\) \(\text{g}\), with the leeway of \(\text{5}\) percent under or \(\text{10}\) percent over. Vishnu is interested in how a well-known, national retailer measures up to this standard. He visited his local branch of the supplier and recorded the masses of \(\text{10}\) different loaves of bread for one week. The results, in grams, are given below:

Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday
\(\text{802.4}\)	\(\text{787.8}\)	\(\text{815.7}\)	\(\text{807.4}\)	\(\text{801.5}\)	\(\text{786.6}\)	\(\text{799.0}\)
\(\text{796.8}\)	\(\text{798.9}\)	\(\text{809.7}\)	\(\text{798.7}\)	\(\text{818.3}\)	\(\text{789.1}\)	\(\text{806.0}\)
\(\text{802.5}\)	\(\text{793.6}\)	\(\text{785.4}\)	\(\text{809.3}\)	\(\text{787.7}\)	\(\text{801.5}\)	\(\text{799.4}\)
\(\text{819.6}\)	\(\text{812.6}\)	\(\text{809.1}\)	\(\text{791.1}\)	\(\text{805.3}\)	\(\text{817.8}\)	\(\text{801.0}\)
\(\text{801.2}\)	\(\text{795.9}\)	\(\text{795.2}\)	\(\text{820.4}\)	\(\text{806.6}\)	\(\text{819.5}\)	\(\text{796.7}\)
\(\text{789.0}\)	\(\text{796.3}\)	\(\text{787.9}\)	\(\text{799.8}\)	\(\text{789.5}\)	\(\text{802.1}\)	\(\text{802.2}\)
\(\text{789.0}\)	\(\text{797.7}\)	\(\text{776.7}\)	\(\text{790.7}\)	\(\text{803.2}\)	\(\text{801.2}\)	\(\text{807.3}\)
\(\text{808.8}\)	\(\text{780.4}\)	\(\text{812.6}\)	\(\text{801.8}\)	\(\text{784.7}\)	\(\text{792.2}\)	\(\text{809.8}\)
\(\text{802.4}\)	\(\text{790.8}\)	\(\text{792.4}\)	\(\text{789.2}\)	\(\text{815.6}\)	\(\text{799.4}\)	\(\text{791.2}\)
\(\text{796.2}\)	\(\text{817.6}\)	\(\text{799.1}\)	\(\text{826.0}\)	\(\text{807.9}\)	\(\text{806.7}\)	\(\text{780.2}\)

1. Is this data set qualitative or quantitative? Explain your answer.

2. Determine the mean, median and mode of the mass of a loaf of bread for each day of the week. Give your answer correct to 1 decimal place.

3. Based on the data, do you think that this supplier is providing bread within the South African regulations?

Qualitative or quantitative?

Since each mass can be represented by a number, the data set is quantitative. Furthermore, since a mass can be any real number, the data are continuous.

Calculate the mean

In each column (for each day of the week), we add up the measurements and divide by the number of measurements, \(\text{10}\).

For Monday, the sum of the measured values is \(\text{8 007.9}\) and so the mean for Monday is

\[\cfrac{\text{8 007.9}}{10} = \text{800.8}\text{ g}\]

In the same way, we can compute the mean for each day of the week. See the table below for the results.

Calculate the median

In each column we sort the numbers from lowest to highest and find the value in the middle. Since there are an even number of measurements (\(\text{10}\)), the median is halfway between the two numbers in the middle.

For Monday, the sorted list of numbers is

\begin{align*} \text{789.0}; \text{789.0}; \text{796.2}; \text{796.7}; \text{801.2}; \\ \text{802.3}; \text{802.3}; \text{802.5}; \text{808.7}; \text{819.6} \end{align*}

The two numbers in the middle are \(\text{801.2}\) and \(\text{802.3}\) and so the median is

\[\cfrac{\text{801.2} + \text{802.3}}{2} = \text{801.8}\text{ g}\]

In the same way, we can compute the median for each day of the week:

Day	Mean	Median
Monday	\(\text{800.8}\) \(\text{g}\)	\(\text{801.8}\) \(\text{g}\)
Tuesday	\(\text{797.2}\) \(\text{g}\)	\(\text{796.1}\) \(\text{g}\)
Wednesday	\(\text{798.4}\) \(\text{g}\)	\(\text{797.2}\) \(\text{g}\)
Thursday	\(\text{803.4}\) \(\text{g}\)	\(\text{800.8}\) \(\text{g}\)
Friday	\(\text{802.0}\) \(\text{g}\)	\(\text{804.3}\) \(\text{g}\)
Saturday	\(\text{801.6}\) \(\text{g}\)	\(\text{801.4}\) \(\text{g}\)
Sunday	\(\text{799.3}\) \(\text{g}\)	\(\text{800.2}\) \(\text{g}\)

From the above calculations we can see that the means and medians are close to one another, but not quite equal. In the next worked example we will see that the mean and median are not always close to each other.

Determine the mode

Since the data are continuous we cannot compute the mode. In the next section we will see how we can group data in order to make it possible to compute an approximation for the mode.

Conclusion: Is the supplier reliable?

From the question, the requirements are that the mass of a loaf of bread be between \(\text{800}\) \(\text{g}\) minus \(\text{5}\%\), which is \(\text{760}\) \(\text{g}\), and plus \(\text{10}\%\), which is \(\text{880}\) \(\text{g}\). Since every one of the measurements made by Vishnu lies within this range and since the means and medians are all close to \(\text{800}\) \(\text{g}\), we can conclude that the supplier is reliable.

The heights of \(\text{10}\) learners are measured in centimetres to obtain the following data set:

\[\{150; 172; 153; 156; 146; 157; 157; 143; 168; 157\}\]

Afterwards, we include one more learner in the group, who is exceptionally tall at \(\text{181}\) \(\text{cm}\).

Compare the mean and median of the heights of the learners before and after the eleventh learner was included.

Calculate the mean of the first \(\text{10}\) learners

\begin{align*} \text{mean } & = \cfrac{150 + 172 + 153 + 156 + 146 + 157 + 157 + 143 + 168 + 157}{10} \\ & = \text{155.9}\text{ cm} \end{align*}

Calculate the mean of all \(\text{11}\) learners

\begin{align*} \text{mean } & = \cfrac{150 + 172 + 153 + 156 + 146 + 157 + 157 + 143 + 168 + 157 + 181}{11} \\ & = \text{158.2}\text{ cm} \end{align*}

From this we see that the average height changes by \(\text{158.2} - \text{155.9} = \text{2.3}\text{ cm}\) when we introduce the outlier value (the tall person) to the data set.

Calculate the median of the first \(\text{10}\) learners

To find the median, we need to sort the data set:

\[\{143; 146; 150; 153; 156; 157; 157; 157; 168; 172\}\]

Since there are an even number of values, \(\text{10}\), the median lies halfway between the fifth and sixth values:

\[\text{median } = \cfrac{156 + 157}{2} = \text{156.5}\text{ cm}\]

Calculate the median of all \(\text{11}\) learners

After adding the tall learner, the sorted data set is

\[\{143; 146; 150; 153; 156; 157; 157; 157; 168; 172; 181\}\]

Now, with \(\text{11}\) values, the median is the sixth value: \(\text{157}\) \(\text{cm}\). So, the median changes by only \(\text{0.5}\) \(\text{cm}\) when we add the outlier value to the data set.

In general, the median is less affected by the addition of outliers to a data set than the mean is. This is important because it is quite common that outliers are measured during an experiment, because of problems with the equipment or unexpected interference.