Sarita wrote down how much of each food item her house used during one whole month. Written out as sentence after sentence, that information is slow to use, so it is set out in a table instead. Each row of the table carries one item and the amount of it that was used, and every amount is written with its unit. A reader can then find any single item at once, without reading through the whole page.

The table now answers a good many questions on its own. The largest amount in the last column is 45 kg of rice, so rice is the item this house uses most. The smallest amount is 1 kg of spices, so spices are used least. The sugar row shows that 4 kg of sugar goes in a month. To find how much food the house used altogether, every amount in the last column is added:

\[ 12 + 45 + 8 + 4 + 2 + 6 + 1 = 78 \]

So this house uses 78 kg of food in a month. Notice what the table did. It did not create any new information. It only arranged the information that was already collected so that a reader can use it.

Any information collected about a subject, by asking, by measuring or by counting, is called data in mathematics. Asking classmates which vegetable they like, measuring the heights of students, noting down the daily wages of workers in a factory: each of these is a collection of data.

When a list holds thirty or forty entries, counting each answer in your head goes wrong very easily. So a mark is made while the counting is going on. Every time an answer appears, one upright stroke is put down in its row. After four strokes, the fifth stroke is drawn across those four, and that makes a bundle of five. Such a bundle is written here as ||||/ . Bundles make the final count quick, because you count the bundles in fives and then add the loose strokes left over.

A shoe shop wrote down the size of every pair of school shoes it sold in one day. This is raw data, because the sizes are written in the order the pairs were sold and in no other order:

• 5, 4, 3, 5, 6, 4, 5, 3, 4, 7
• 5, 5, 4, 6, 3, 5, 4, 7, 5, 4
• 3, 6, 5, 4, 5, 6, 4, 3, 7, 5

To put this into a frequency table, work through the following order.

• Write the different sizes in the first column, from the smallest to the largest.
• Go through the list in the order it was written, and for each entry put one stroke in the row of that size.
• Make bundles as you go, crossing every fifth stroke over the four before it.
• Count the strokes in each row and write that number in the frequency column.
• Add the frequency column, and check that the total is equal to the number of entries in the data.

The last step is the check, and it is done by adding the frequency column:

\[ 5 + 8 + 10 + 4 + 3 = 30 \]

The list held 30 entries and the frequency column also totals 30, so the counting is correct. The table now gives these answers straight away.

• (a) The size sold most is 5, and 10 pairs of it were sold.
• (b) The size sold least is 7, and only 3 pairs of it were sold.
• (c) The shop sold 30 pairs of shoes altogether that day.

A table gives the numbers, but a picture is better when the numbers have to be compared by eye. For that, upright rectangles are used, and the height of each rectangle stands for the frequency of that item. A taller rectangle simply means a larger number.

Four things must be looked after while a bar diagram is being drawn.

• The X axis and the Y axis must both be drawn clearly and named.
• Every bar diagram must carry a title.
• Every bar must have the same width.
• The gap between any two bars must be the same.

On tree planting day the students of classes 6 to 10 of a school planted the following numbers of trees. The steps below show this data as a simple bar diagram.

Step 1: The two axes are drawn on graph paper and named. The classes go along the X axis and the number of trees goes up the Y axis.

Step 2: The largest value is 45, so one small square on the Y axis is taken to stand for 5 trees, and the axis is marked in equal jumps of 5, 10, 15 and so on.

Step 3: The tallest bar belongs to class 7. Dividing 45 trees by the scale gives its height in small squares:

\[ 45 \div 5 = 9 \]

Step 4: Class 6 planted 30 trees, so the height of its bar is:

\[ 30 \div 5 = 6 \]

Step 5: The heights of the other three bars are found in the same way:

\[ 25 \div 5 = 5 \]

\[ 40 \div 5 = 8 \]

\[ 35 \div 5 = 7 \]

Step 6: All five bars are drawn with the same width and with equal gaps between them, the bars are coloured, and the title is written above the diagram.

Once the diagram is finished, adding the values of all the bars and checking them against the data is a good habit:

\[ 30 + 45 + 25 + 40 + 35 = 175 \]

So 175 trees were planted in that school altogether.

To read a finished bar diagram, look across from the top of a bar to the Y axis, and the number written there is the value of that bar. Below, the population of a town over six years is shown in lakh.

The tallest bar is the one for 2075, so the population was largest in that year at 100 lakh. The bar for 2072 reaches 50, so the population that year was 50 lakh. To find the rise from one year to the next, the two values are subtracted. The rise from 2073 to 2074, for example, is:

\[ 90 - 65 = 25 \]

In the same way, the rise from 2070 to 2071 is:

\[ 40 - 35 = 5 \]

Comparing the rise for every year, this 5 lakh is the smallest of them, so the smallest rise came in 2071 and it was 5 lakh. Writing the values read off the bars back into a table gives the frequency table again, which shows that you can travel both ways, from table to diagram and from diagram to table.