Numerical Data Displays and Spread: Practice

Every dot is one value, so count all the dots, whichever value they sit above.

$3 + 5 + 2 = 10$

The $3$ counts how many different values have dots, and the $5$ is only the tallest stack.

A dot plot puts one dot above the number line for each value, so each value can be read back from the picture.

$\text{one dot} = \text{one value}$

A histogram only counts how many values fall in each interval, and a box plot shows just five numbers. A circle graph shows shares of a whole, not the values themselves.

The five-number summary lists, in order, the minimum, the first quartile, the median, the third quartile, and the maximum.

$\text{min},\ Q_1,\ \text{median},\ Q_3,\ \text{max}$

The mean is a useful center, but it is not one of the five.

There are $7$ values, an odd count, so the median $7$ is left out of both halves. The lower half is the three values before it, $2, 4, 5$.

$Q_1 = \text{median of } 2, 4, 5 = 4$

With $7$ values the median $7$ is left out, so the upper half is the three values after it, $8, 10, 12$.

$Q_3 = \text{median of } 8, 10, 12 = 10$

The maximum, $12$, is the end of the data, not its third quartile.

The interquartile range is the distance from the first quartile to the third quartile, so subtract.

$\text{IQR} = Q_3 - Q_1 = 20 - 12 = 8$

Adding the quartiles, $32$, does not measure a distance at all.

A box plot draws the five-number summary. The box runs from $Q_1$ to $Q_3$, and the line inside it marks the middle number of the five.

$Q_1 \le \text{median} \le Q_3$

The mean and the mode are not part of a box plot, and $Q_1$ is the left edge of the box.

A histogram's horizontal axis is a number line cut into equal intervals. Each interval begins where the one before it ends, so the bars meet.

$0 \text{ to } 9,\quad 10 \text{ to } 19,\quad 20 \text{ to } 29$

A bar graph's bars stand for separate categories, which have no fixed order, so they are drawn apart.

In a five-number summary the second number is $Q_1$ and the fourth is $Q_3$.

$\text{IQR} = 17 - 9 = 8$

The range is $25 - 4 = 21$, which uses the two ends instead of the quartiles.

Each whole number belongs to exactly one interval. The interval $5$ to $9$ stops at $9$, and the next one starts at $10$.

$10 \le 10 \le 14$

So the value $10$ is counted once, in the $10$ to $14$ bar.

The box runs from $Q_1$ to $Q_3$, which cut the sorted list at about one quarter and three quarters of the way along. With no repeated values, the values between them are about the middle half of the list.

$\tfrac34 - \tfrac14 = \tfrac12$

Repeated values can change this: a value repeated at a quartile sits on the edge of the box, and then the box can hold many more than half.

The data is sorted and has $7$ values, so the median is the $4$th value, $6$, and it is left out of both halves.

The lower half $1, 3, 4$ has middle value $3$, and the upper half $8, 9, 10$ has middle value $9$.

$1,\ 3,\ 6,\ 9,\ 10$

Keeping the $6$ in both halves would give $3.5$ and $8.5$, which is a different method from this course's.