Scatter Plots, Histograms and Distribution Curves

Reading a dot's x and y, the highest or leftmost dot, differences, averages.

In a scatter plot every dot shows two values of one item (one student, one shop, one day). One value is on the horizontal axis (x-axis) and the other on the vertical axis (y-axis). The dot's position is fixed by the pair $(x,y)$, called its coordinates: always x (horizontal) first, then y (vertical). In the figure every shop is a dot and the letter is its name.

1. Read both axis titles and units (₹ thousand? lakh? hours?) and check whether the scale starts at zero.
2. From the dot go down to the x-axis and read x; then go left to the y-axis and read y.
3. To compare two dots write both values separately, then subtract, divide or add.

Recognising the relationship between two variables.

Correlation tells how one quantity ($y$) changes when the other ($x$) changes. Positive correlation: y rises as x rises (dots climb left to right, e.g. study hours and marks). Negative correlation: y falls as x rises (dots descend, e.g. temperature and sales of hot tea). No correlation: the dots are scattered at random (e.g. roll number and marks).

Strength: the more tightly the dots hug a straight line, the stronger the correlation; the more scattered, the weaker. In numbers it is given by the correlation coefficient $r$.

Here: $r$ = correlation coefficient; its sign gives the direction and its size (distance from 0) gives the strength.

1. Look left to right: do the dots climb, descend or scatter?
2. Then check tightness: close to a line = strong, spread out = weak.
3. If r is given, read the direction from its sign and the strength from its distance from 0.

Using the trend line to estimate.

When the dots lie roughly around a straight line, we draw one straight line through them. It is called the line of best fit (trend line). It is drawn so that about as many dots lie above it as below, and the dots stay near it. With the line drawn we can read an estimate of y for any x: go up from x to the line, then left to the y-axis.

1. Take two clean points on the line (where it crosses grid lines) and read their (x, y).
2. Find the slope $m=\frac{y_2-y_1}{x_2-x_1}$; then $c=y_1-m{x}_1$.
3. For an estimate put x in $y=c+mx$ (or read it straight off the figure).

Spotting clusters and outliers; their effect on the average.

A cluster is a group of dots lying close together and separated from other groups by a gap. An outlier is a dot that belongs to no group and lies far from the rest. It may be a wrong entry or a genuinely unusual event, so do not delete it without thinking.

• Effect on the average: a single outlier pulls the average toward itself, so the average with it differs from the average without it.
• Effect on correlation: one outlier can change r a lot, because it lies far from the line.
• Meaning of clusters: separate clusters often show different kinds of items (here short-distance and long-distance deliveries).

1. First count the groups separated by gaps.
2. A dot in no group is an outlier; read its coordinates.
3. For an average, write the sum and the count both with and without the outlier.

Counting dots by a given line or by four parts and finding percents.

Questions often give a vertical line $x=a$ and a horizontal line $y=b$. They cut the plot into four parts: top-right ($x>a$, $y>b$), top-left ($x<a$, $y>b$), bottom-right ($x>a$, $y<b$), bottom-left ($x<a$, $y<b$). Sometimes a slanted line (such as the best-fit line) is given and you must count the dots above or below it. Above = a larger y than the line has at the same x.

1. Write down the conditions (such as x > a and y > b).
2. Check each dot one by one and tick it off as you count.
3. For percent, divide the count by the total number of dots and multiply by 100.

Finding ratios, percents, averages and per-unit rates from a scatter plot.

Most questions ask you to read a scatter plot like a table. The safest way: first make a small table from the figure (dot, x, y), then answer everything from the table. Ratios, percents, averages and $y÷x$ (the rate per unit) then become ordinary table questions.

Here: $x$ = experience (years), $y$ = monthly salary (₹ thousand), $y÷x$ = salary per year of experience.

1. Write each dot's x and y to make the table.
2. Add a new column for what is asked ($y÷x$, a difference, etc.).
3. Identify the base for percent: in "A is what % more than B" the base is B.

The slips that cost marks.

• Axis not starting at zero: then never take the ratio of heights as the ratio of values. Both plots below show the same data; the left starts at 0 and the right at 40, so the differences look bigger on the right.
• Different scales: when comparing two figures read each axis scale separately; slopes that look alike do not mean the same thing.
• Hidden dots: items with equal (x, y) fall on top of each other, so the visible dots can be fewer than the items.
• Order and unit: coordinates are x first; put the axis unit (thousand, lakh) in the answer.

Class intervals, frequency, total, percent, modal class.

How to read a histogram when the classes are not equally wide.

The three averages of grouped data.

The difference between the two and bars of cumulative frequency.

Drawing and reading a polygon; comparing two polygons.

Drawing and reading an ogive.

Reading x from the heights N/2, N/4 and 3N/4.

Telling the order of the three averages from the shape of the distribution.

The area under the curve gives percents; the percents of the three bands.

Comparing different scales and telling a position.

Reading a box plot and comparing two groups.

Checking which value is an outlier.

Median, mode and counts from the plot; comparing two classes.

Total, mode, median and mean from the dots.

The boundary rule from open and filled dots.

The same four calculations from any graph: change, ratio, average, the year of highest growth.

Testing the truth of statements against a graph.

A pie with a bar chart, and a histogram with an ogive.

Each graph: what to read and the key move.

Five exam-style questions on each graph.

One slip per graph.