Descriptive statistics: data, tables and graphs

Answer: $7$

Add the frequencies up to and including $1\mathclose{:}$ $4 + 3 = 7\mathclose{.}$

Common mistakes, and the answer each one gives:

• Left out the value $1$ itself → $4$
• Gave the frequency of $1$ alone → $3$

Answer: $15$

Add the bars from $2$ upwards: $6 + 2 + 7 = 15\mathclose{.}$

Common mistakes, and the answer each one gives:

• Left out the bar for $2$ itself → $9$
• Counted the bars instead of adding their heights → $3$

Answer: $\text{mode}$

Nominal values cannot be ordered or added: only the mode is meaningful.

Common mistakes, and the answer each one gives:

• Used the median, which needs an order → $\text{median}$

Answer: $1$

Each value counts as often as its frequency: the sum is $0 \cdot 8 + 1 \cdot 2 + 2 \cdot 3 + 3 \cdot 1 + 4 \cdot 1 = 15\mathclose{,}$ over $15$ observations: $\dfrac{15}{15} = 1\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took the mean of the values, ignoring the frequencies → $2$
• Divided by the number of different values → $3$

Answer: $\frac{17}{9}$

Read the frequencies off the bars, $6, 7, 2, 8, 4\mathclose{;}$ then the sum is $0 \cdot 6 + 1 \cdot 7 + 2 \cdot 2 + 3 \cdot 8 + 4 \cdot 4 = 51$ over $27$ observations: $\frac{17}{9}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took the mean of the bar heights → $5.4$
• Divided by the number of bars → $10.2$

Answer: $\text{ratio}$

It is ratio: the steps are equal and 0 means none at all, so ratios mean something.

Common mistakes, and the answer each one gives:

• Missed that 0 seconds is a true zero → $\text{interval}$

Answer: $2.5$

There are $18$ observations, so the median is the average of the 9th and 10th in order. Counting through the frequencies in order of value, that is $2.5\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took the middle value of the table's top row → $3$
• Gave the value with the largest frequency → $1$

Answer: $1$

The mode is the value with the tallest bar: $1\mathclose{,}$ which occurs $8$ times.

Common mistakes, and the answer each one gives:

• Gave the height of the tallest bar instead of its value → $8$
• Gave the largest value → $5$

Answer: $\text{qualitative}$

It is qualitative: its values are categories, not numbers to calculate with.

Common mistakes, and the answer each one gives:

• Took the colour names for a measured quantity → $\text{quantitative and discrete}$

Answer: $\dfrac{1}{32}$

The relative frequency is the frequency divided by the total number of observations, $1 + 6 + 7 + 9 + 9 = 32\mathclose{:}$ $\dfrac{1}{32}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Divided by the number of different values instead of the total → $\dfrac{1}{5}$
• Gave the frequency itself → $1$

Answer: $30$

Add the heights of the bars: $5 + 3 + 8 + 7 + 7 = 30\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added the values under the bars instead of their heights → $10$
• Counted the bars → $5$

Answer: $\text{mean}$

Interval values can be added and divided, so the mean is meaningful — a true zero is only needed for ratios.

Common mistakes, and the answer each one gives:

• Thought a mean needs a true zero → $\text{median}$

Answer: $16$

Add the frequencies up to and including $3\mathclose{:}$ $6 + 2 + 3 + 5 = 16\mathclose{.}$

Common mistakes, and the answer each one gives:

• Left out the value $3$ itself → $11$
• Gave the frequency of $3$ alone → $5$

Answer: $19$

Add the bars from $1$ upwards: $6 + 7 + 5 + 1 = 19\mathclose{.}$

Common mistakes, and the answer each one gives:

• Left out the bar for $1$ itself → $13$
• Counted the bars instead of adding their heights → $4$

Answer: $\text{nominal}$

It is nominal: the values are only names: they cannot be put in a meaningful order.

Common mistakes, and the answer each one gives:

• Saw an order in the colours → $\text{ordinal}$

Answer: $\frac{7}{3}$

Each value counts as often as its frequency: the sum is $0 \cdot 5 + 1 \cdot 2 + 2 \cdot 2 + 3 \cdot 5 + 4 \cdot 7 = 49\mathclose{,}$ over $21$ observations: $\dfrac{49}{21} = \frac{7}{3}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took the mean of the values, ignoring the frequencies → $2$
• Divided by the number of different values → $9.8$

Answer: $\frac{79}{26}$

Read the frequencies off the bars, $6, 3, 5, 8, 4\mathclose{;}$ then the sum is $1 \cdot 6 + 2 \cdot 3 + 3 \cdot 5 + 4 \cdot 8 + 5 \cdot 4 = 79$ over $26$ observations: $\frac{79}{26}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took the mean of the bar heights → $5.2$
• Divided by the number of bars → $15.8$

Answer: $2.5$

There are $34$ observations, so the median is the average of the 17th and 18th in order. Counting through the frequencies in order of value, that is $2.5\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took the middle value of the table's top row → $3$
• Gave the value with the largest frequency → $2$

Answer: $\text{quantitative and discrete}$

It is quantitative and discrete: it is a count: whole numbers, with nothing in between.

Common mistakes, and the answer each one gives:

• Took a count to be continuous → $\text{quantitative and continuous}$

Answer: $2$

The mode is the value with the tallest bar: $2\mathclose{,}$ which occurs $8$ times.

Common mistakes, and the answer each one gives:

• Gave the height of the tallest bar instead of its value → $8$
• Gave the largest value → $5$

Answer: $\dfrac{6}{25}$

The relative frequency is the frequency divided by the total number of observations, $3 + 5 + 3 + 6 + 8 = 25\mathclose{:}$ $\dfrac{6}{25}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Divided by the number of different values instead of the total → $\dfrac{6}{5}$
• Gave the frequency itself → $6$

Answer: $\text{median}$

Ordinal values can be ordered but not added: the median is meaningful, the mean is not.

Common mistakes, and the answer each one gives:

• Used the mean, which needs equal steps → $\text{mean}$
• Used the mode, which throws away the order → $\text{mode}$

Answer: $26$

Add the heights of the bars: $8 + 6 + 3 + 3 + 6 = 26\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added the values under the bars instead of their heights → $10$
• Counted the bars → $5$

Answer: $11$

Add the frequencies up to and including $2\mathclose{:}$ $7 + 4 = 11\mathclose{.}$

Common mistakes, and the answer each one gives:

• Left out the value $2$ itself → $7$
• Gave the frequency of $2$ alone → $4$

Answer: $\text{interval}$

It is interval: the steps are equal, but the zero is not a true zero, so ratios mean nothing (20 is not twice 10).

Common mistakes, and the answer each one gives:

• Took 0 degrees Fahrenheit to mean no temperature at all → $\text{ratio}$

Answer: $12$

Add the bars from $3$ upwards: $4 + 8 = 12\mathclose{.}$

Common mistakes, and the answer each one gives:

• Left out the bar for $3$ itself → $8$
• Counted the bars instead of adding their heights → $2$

Answer: $\frac{30}{17}$

Each value counts as often as its frequency: the sum is $0 \cdot 5 + 1 \cdot 1 + 2 \cdot 6 + 3 \cdot 3 + 4 \cdot 2 = 30\mathclose{,}$ over $17$ observations: $\dfrac{30}{17} = \frac{30}{17}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took the mean of the values, ignoring the frequencies → $2$
• Divided by the number of different values → $6$

Answer: $\text{quantitative and continuous}$

It is quantitative and continuous: it is measured, and can take any value in a range.

Common mistakes, and the answer each one gives:

• Took a measurement to be a count → $\text{quantitative and discrete}$

Answer: $\frac{63}{22}$

Read the frequencies off the bars, $1, 1, 5, 8, 7\mathclose{;}$ then the sum is $0 \cdot 1 + 1 \cdot 1 + 2 \cdot 5 + 3 \cdot 8 + 4 \cdot 7 = 63$ over $22$ observations: $\frac{63}{22}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took the mean of the bar heights → $4.4$
• Divided by the number of bars → $12.6$

Answer: $1$

There are $32$ observations, so the median is the average of the 16th and 17th in order. Counting through the frequencies in order of value, that is $1\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took the middle value of the table's top row → $2$
• Gave the value with the largest frequency → $0$