A cross-tabulation table (クロス集計表) places one item along the rows and another along the columns, and counts in each cell the number of records that meet both conditions.
Group
Eats breakfast
Skips breakfast
Total
Students
18
12
30
Working adults
25
15
40
Total
43
27
70
For example, put the group (student or working adult) on the rows and whether people eat breakfast on the columns, then count the people in each combination. Adding total cells at the end of the rows and columns lets you check the overall count and the breakdown at once. Dividing each cell by its row total turns it into a proportion, so groups of different sizes can still be compared.
AnalogyJust as a school timetable fixes one class in the cell where a weekday meets a period, each cell here holds one count where a row condition meets a column condition.
How to rememberCross = two lines meeting like a plus sign.
Exam points
ExamWhen reading a cross-tabulation table (クロス集計表), first decide whether you compare within a row or within a column. The cell with the largest count can sit in a different place in each row.
ExamDividing each cell by the grand total of the table gives its share of the whole (構成比). This differs from a proportion based on the row total because the divisor is different.
A contingency table (分割表) is the statistics name for a cross-tabulation table; both names refer to the same table.
It refers to a table that splits all the data by the categories of two items and lists the frequency (count) for each combination. This name is used, for example, when testing statistically whether two items are related.
A mosaic plot (モザイク図) turns the counts of a cross-tabulation table into rectangle areas to show the proportions within each group.
Counts in the cross-tabulation table
Split the width by the shares of one item
Split each width vertically by the shares of the other item
Tiles whose areas are proportional to the counts
Each of the 4 resulting tiles has an area proportional to the number of people in its cell.
4 / 4
First, set the widths according to the share of each category of one item. Next, cut each width vertically according to the shares of the other item. Because each tile's area is proportional to its cell's count, one figure shows both the size of each group and how their breakdowns differ.
AnalogyIt is like snapping a chocolate bar vertically by the share of people, then snapping each piece horizontally by its breakdown, so the size of each small piece stands for a number of people.
How to rememberLike a mosaic artwork, rectangular tiles of different sizes fit together into one picture.
Exam points
Mix-upA 100% stacked bar chart (帯グラフ) divides bands of equal length by their breakdown, and every band has the same width. A mosaic plot (モザイク図) also makes each band's width proportional to the count of its group.
Cross-tabulation table vs. Mosaic plot
Aspect
Cross-tabulation table
Mosaic plot
Key differenceHow it is shown
Lists the counts as numbers
Turns the counts into rectangle areas
Easy to read
The exact count in each cell
The overall picture of group sizes and their breakdowns
A correlation coefficient matrix (相関係数行列) lists several variables in the same order along the rows and columns, and writes the correlation coefficient of each pair in the cell where they meet.
Variable
Temperature
Store visitors
Ice cream sales
Temperature
1.00
0.35
0.82
Store visitors
0.35
1.00
0.48
Ice cream sales
0.82
0.48
1.00
A correlation coefficient shows the strength and direction of the linear relationship between two variables, and ranges from −1 to 1. The diagonal cells, where a variable meets itself, are 1, and the same values mirror each other across the diagonal. Even with many variables, scanning the off-diagonal cells for values close to 1 or −1, whatever the sign, finds the strongly related pairs at once.
AnalogyIt has the same shape as a city distance chart with the same cities on the rows and columns: A to B equals B to A, so identical numbers face each other across the diagonal.
Exam points
ExamThe closer a positive correlation coefficient (相関係数) is to 1, the stronger the tendency for one variable to be large when the other is large.
A scatter plot matrix (散布図行列) places a scatter plot for each pair of variables in a grid that lists the variables along its rows and columns.
Variable
Temperature
Store visitors
Ice cream sales
Temperature
Distribution of temperature
Temperature and visitors scatter plot
Temperature and ice cream sales scatter plot
Store visitors
Visitors and temperature scatter plot
Distribution of visitors
Visitors and ice cream sales scatter plot
Ice cream sales
Ice cream sales and temperature scatter plot
Ice cream sales and visitors scatter plot
Distribution of ice cream sales
Each cell holds a scatter plot that uses the values of its row variable and its column variable as the two axes. The diagonal cells pair a variable with itself, so they can hold the variable name or that variable's own distribution. Points rising to the upper right show a positive correlation and points falling to the lower right show a negative one, which you can confirm by eye.
AnalogyIt is like a seating chart in roster order with a photo of a pair of people pinned to every seat, so every possible pair can be seen at a glance.
Exam points
Mix-upA correlation coefficient matrix (相関係数行列) sums up each relationship's strength in a single number, while a scatter plot matrix (散布図行列) shows how the points themselves are arranged. Curved relationships and outliers are easier to spot in a scatter plot matrix.
Correlation coefficient matrix vs. Scatter plot matrix
Aspect
Correlation coefficient matrix
Scatter plot matrix
Layout
The same variables in the same order on rows and columns
The same variables in the same order on rows and columns
Key differenceCell contents
Correlation coefficients (numbers)
Scatter plots (point patterns)
What it reveals
Strength and direction of linear relationships
Point patterns, including curved relationships and outliers
A combination chart (複合グラフ) overlays different kinds of charts, such as a bar chart and a line chart, in one figure.
For example, showing amounts as bars and ratios or changes over time as a line lets you compare amounts and changes on the same time axis. A moving total is the sum over a fixed period, recalculated as the period shifts forward one step at a time, and its line can be drawn over bars of the original values.
AnalogyLayering land-use colors and railway lines onto one map lets you compare information drawn in different ways on the same base.
How to rememberCombination = putting two or more things together into one.
Exam points
ExamEven in a chart that overlays bars and a line, the profit margin (利益率) is profit divided by sales (売上高), compared at each point in time. A growth rate (伸び率) is likewise the increase divided by the value at the base point.
ExamEven if costs keep rising, the ratio of costs to sales falls as long as sales grow at a faster rate.
ExamWhen a moving total (移動合計) covers 12 months, every point's total includes each month of the year exactly once, so seasonal ups and downs are smoothed out and the line shows the long-term trend of the original values.
A dual-axis chart (2軸グラフ) has a vertical axis with its own scale on each side, left and right, and overlays the series in one figure with each series tied to one of the two axes.
One vertical axis
Two vertical axes, one on each side
Scale
Shared by all series
Separate for the left and right axes
Series with very different magnitudes
The smaller series hugs the bottom, so its changes are hard to see
Each series shows its changes on the scale of its own axis
For example, the left axis may show amounts (in 100 million yen) and the right axis quantities (in thousands of units), with series of the same unit grouped on the same side. Because one axis can carry several series, first check the legend or the axis titles to see which axis each series belongs to.
Exam points
Mix-upThe left and right axes use separate scales, so when a left-axis line crosses a right-axis line, it does not mean the two values are equal at that point.
Combination chart vs. Dual-axis chart
Aspect
Combination chart
Dual-axis chart
What is overlaid
Different kinds of charts
Series with different units or magnitudes (one axis may carry several)
Key differenceVertical axes
Either one or two
Two, left and right, with different scales
Key points
A cross-tabulation table (a contingency table in statistics) counts each combination of two items, and its trends are read together with the totals and proportions.
A mosaic plot turns the counts of a cross-tabulation table into tile areas, showing group sizes and breakdowns in one figure.
In a correlation coefficient matrix, the diagonal is 1, the same values mirror each other across it, and values from −1 to 1 show the strength and direction of each relationship.
A combination chart overlays different kinds of charts, such as bars and a line, so amounts and changes can be compared in one figure.
A dual-axis chart has separate scales on its left and right vertical axes, so series of very different magnitudes can each be followed in one figure.