Bar Graphs and Box Plots - ISEE Upper Level: Mathematics Achievement

$24$. Subtract $Q_1$ from $Q_3$ to determine the interquartile range, measuring central data dispersion.

The minimum to $Q_1$ and $Q_3$ to maximum. Whiskers extend from the box edges to the extreme values, representing the lower and upper quartiles of the data.

$Q_3 - Q_1$. IQR measures the dispersion of the middle 50% of the data, providing a robust indicator of variability resistant to outliers.

$50%$. By definition, the interquartile range encompasses exactly half of the ordered data points in the set.

$50%$. The median divides the ordered data such that half the values are less than or equal to it.

$25%$. The first quartile $Q_1$ marks the boundary below which the lowest quarter of the ordered data falls.

$25%$. The third quartile $Q_3$ delineates the point above which the highest quarter of the ordered data lies.

The median. Box plots visually display the median as the central line within the box, representing the data's center.

The median. In skewed distributions, the median provides a better central tendency measure as it is less influenced by extreme values than the mean.

The value (frequency or amount) for that category. In a bar graph, each bar's height directly corresponds to the magnitude of the data for its specific category, enabling visual comparison.

The categories (groups) being compared. The horizontal axis in a vertical bar graph labels distinct categories or groups, facilitating comparison of their respective values.

The numerical scale for the values (counts or amounts). The vertical axis provides a scaled measurement axis for quantifying the frequency or amount associated with each category.

Bars have equal width and are separated by gaps. Uniform bar widths ensure equitable visual representation, while gaps clearly separate discrete categories for accurate interpretation.

$\text{max} - \text{min}$. The range quantifies the total spread of the data by subtracting the smallest value from the largest in the set.

$35$. Subtract the minimum from the maximum to compute the total spread of the data as shown on the plot.

$9$. Calculate IQR by subtracting the second value ($Q_1$) from the fourth value ($Q_3$) in the summary.

$\frac{5 + 8}{2} = 6.5$. For an even-numbered ordered data set, the median is the average of the two middle values.

$7$. For an odd-numbered ordered data set, the median is the central value in the list.

$12$. In the five-number summary, the third value represents the median of the data set.

$6$. The second value in the five-number summary corresponds to the first quartile $Q_1$.

$21$. The fourth value in the five-number summary indicates the third quartile $Q_3$.

Minimum, $Q_1$, median, $Q_3$, maximum. These values summarize the data distribution by dividing the ordered set into four equal parts for box plot construction.

The interval from $Q_1$ to $Q_3$ (the middle $50%$). The box encloses the central half of the ordered data, illustrating the spread of the middle two quartiles.

The median ($Q_2$) of the data. This internal line marks the midpoint of the ordered data set, dividing it into two equal halves.