Distinguish aggregation, which condenses data into a few values, from the operations that put data in order and assign each item a rank.
Arrange the four levels of measurement from the one whose values carry the least information to the one carrying the most, and explain which calculations make sense at each.
Decide whether a set of data is on an interval scale or a ratio scale by looking at what 0 represents.
Data aggregationデータの集計 is the operation of combining individual records to suit a purpose and working out values such as the total, the count and the average.
Branch
Number of transactions
Total sales (10,000 yen)
East branch
3
45
West branch
2
30
All branches
5
75
Record-by-record data makes the overall trend hard to see, so aggregation condenses it into a handful of numbers. You can also split the data into groups by an item such as region or month and work out a subtotal for each group. In spreadsheet software, the functions that add up values or count entries carry out this operation.
AnalogyIt is like adding up a month of household spending by category and listing how much went on food and how much on transport.
How to rememberThe Japanese word 集計 joins 集 (gather) and 計 (measure), naming the two steps of the job: collect the records, then compute the figures.
Exam points
Mix-upAggregation集計 condenses many records into a few representative values, a different goal from sorting 並べ替え, which only changes the order of the data.
Data sortingデータの並べ替え (sortソート) is the operation of arranging data in order by a criterion such as the size of a number or which date comes first.
Arranging from smallest to largest is ascending order昇順, and the reverse is descending order降順. The item that decides the order is called the keyキー, and records that tie on the first key can be ordered further by a second key. Sorting leaves every value unchanged and only moves records to new positions.
AnalogyRearranging the books on a shelf by author name does not change what is inside a single book.
How to rememberAscending means climbing, as values do when they go up a staircase, and descending means walking back down it.
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Mix-upAscending order昇順 starts from the smallest value, so for dates the oldest date comes first. Listing from the newest date is descending order降順.
Rankingランキング means giving each record a rank such as 1st or 2nd, based on the result of sorting by some criterion.
Original data
Sort by the criterion value
Assign ranks
Ranks 1, 2 and so on are assigned in the order the records now stand.
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Whether the largest or the smallest value takes first place depends on the purpose. You also need to decide in advance how to treat records with equal values; one approach gives them the same rank. Spreadsheet software provides a function that returns the rank of a value within a specified range.
AnalogyWhen a marathon records places in the order runners cross the line, neighboring places differ by exactly one whether the runner ahead was a few seconds or several minutes faster.
How to rememberRank also means a level or grade, as in a military rank.
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Mix-upA difference in rank 順位 says nothing about how large the difference in the original values is. The gap between 1st and 2nd need not equal the gap between 2nd and 3rd.
Data Sorting vs. Ranking
Aspect
Data Sorting
Ranking
Key differenceWhat the operation adds
Nothing (only the positions of the data change)
Each record gains a rank as a new value
Original values
Unchanged
Unchanged (the rank is attached as a separate value)
A scale尺度 (level of measurement尺度水準) is a type of data defined by what its values represent, and the scale decides whether calculations such as comparing sizes or taking an average are meaningful.
Nominal scale: can be told apart
Ordinal scale: has an order
Interval scale: equal intervals
Ratio scale: ratios make sense
On a ratio scale, 0 also stands for no quantity, so ratios have meaning too.
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There are four levels: the nominal, ordinal, interval and ratio scales. In that order, each level adds one more property to the values, and the range of meaningful calculations grows with it.
How to rememberThe Japanese word 尺度 originally meant a ruler for measuring length, and different rulers can measure different things.
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ExamWhich scale尺度 applies depends on how far the values go along the properties of distinction, order, equal intervals and a 0 that means no quantity.
A nominal scale名義尺度 is a scale whose values are used only to distinguish and classify things, with no meaning of size or order.
Examples include gender, home region and product category codes. Even when they are written as numbers, those numbers are only labels for telling things apart, like name tags, so adding or averaging them gives nothing meaningful. Meaningful operations include checking whether two values match and counting the entries in each category.
AnalogyThe numbers on station coin lockers tell you which locker is which; locker 10 does not hold twice as much luggage as locker 5.
How to rememberNominal comes from the Latin nomen, meaning name: names can be told apart, but no name is larger than another.
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Mix-upData written as numbers is still on a nominal scale名義尺度 if the numbers are only category labels. Being numeric is not a reason to assume order or size.
An ordinal scale順序尺度 is a scale in which the order of values, larger or smaller, higher or lower, has meaning, but the intervals between values are not necessarily equal.
Survey ratings such as satisfied, somewhat satisfied and neutral, and finishing positions in a race, are examples. You can compare which comes first, but nothing says the gaps between neighboring levels are equal, so differences and averages are hard to interpret. The usual representative value is the one in the middle when the values are lined up in order, called the median (中央値).
AnalogyWith hotel star ratings, nobody guarantees that the step from three stars to four is as big as the step from one star to two.
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Mix-upData whose values have been replaced by ranks is on an ordinal scale順序尺度. Even if the original values were on a ratio scale 比例尺度, turning them into ranks throws away the information about how large the differences are.
An interval scale間隔尺度 is a scale with equal intervals between its marks, so the difference between values has meaning, but 0 does not mean there is no quantity.
Temperature in degrees Celsius and calendar years are examples. Because 0 ℃ is not a state with no temperature but simply the chosen starting point of the scale, 20 ℃ cannot be called twice as hot as 10 ℃. Addition, subtraction and averages are meaningful, but ratios (how many times larger) are not.
AnalogyIf everyone is measured while standing on the same platform of a height gauge, every reading is off by the platform's height, yet the difference between any two people stays the same. The scale keeps differences but not ratios.
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ExamClock time is on an interval scale間隔尺度, because 0 o'clock does not mean an absence of time. You can find the difference between two times, but a ratio of two times has no meaning.
A ratio scale比例尺度 (ratio-level scale比率尺度) is a scale that, besides having equal intervals, has a 0 meaning no quantity, so the ratio of values is meaningful too.
Length, weight and amounts of money are examples. Because 0 is fixed as the state of having none, 60 kg can be called twice 30 kg, and the results of multiplication and division are meaningful. Of the four scales, its values carry the most information.
AnalogyA wallet holding 0 yen really has no money in it, and 2000 yen lets you spend exactly twice as much as 1000 yen.
How to rememberThe name comes from ratio: this is the scale on which a ratio of two values makes sense.
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ExamLength and area are quantities on a ratio scale比例尺度, so they can be compared as ratios. For figures of similar shape, the ratio of their areas is the square of the ratio of their lengths.
Moving from nominal to ratio, each scale尺度 level carries more information, and a higher level supports every calculation that a lower level allows.
Which scale applies is decided by what the value represents, not by whether it happens to be written as a number.
The meaningful representative value also depends on the scale: the most common category (the mode 最頻値) for a nominal scale 名義尺度, the median 中央値 for an ordinal scale 順序尺度, and the mean for an interval scale 間隔尺度 or higher.