5.1 Of Questions and Statements

5.1 Of Questions and Statements

Suppose your teacher tells you that she is meeting two of their childhood friends that evening. One friend is 5 feet tall and the other is 6 feet tall. What is your guess about each friend's gender based on this information?

You might have guessed that the 5-foot-tall person is a woman and the 6-foot-tall person is a man. There is a good chance you are right. But experience tells us that 5-foot-tall men and 6-foot-tall women are rare. We have seen that, more often, men are taller than women.

The above is a simple example of statistical thinking^?. We regularly come across statements like —

We call these statistical statements. Simply put, a statistical statement is a claim or summary about some phenomenon, expressed in terms of numerical values, proportions, probabilities, or predictions.

A statistical question is one that can be answered by collecting data. For example, "How tall are Grade 7 students in our school?" is a statistical question. We expect that not all Grade 7 students have the same height, but we can collect data, analyse it, and make conclusions about the heights that do occur. The question "Typically, are onions costlier in Yahapur or Wahapur?" is also a statistical question. Prices can vary over time. Therefore, answering this question requires us to look at data, analyse it, and come to conclusions making suitable statistical statements.

The term statistics refers to the study of collecting, organising, analysing, interpreting, and presenting data. In this chapter, we shall encounter some statistical questions and learn how analysing data and graphs can help answer them.

5.2 Representative Values

The runs scored by Shubman and Yashasvi in a cricket series are given in the table below. Who do you think performed better?

Shreyas says, "Both their performances are similar since Yashasvi scored more in the first and second matches, whereas Shubman scored more in the third and fourth."

Vaishnavi says, "I think Shubman performed better because he scored the highest number of runs in a match — 90!"

Shreyas says, "No! Yashasvi batted better since the total number of runs he made is 175, while Shubman made only 128."

Vaishnavi says, "Oh! Also, Yashasvi's batting is more consistent — the difference between his maximum score and minimum score is lower."

The table below shows the runs scored by these two players in another series. Who do you think performed better in this series?

Vaishnavi says, "Here, Shubman performed better. His total is 110 runs, while Yashasvi's total is 96 runs."

Can a Single Number Act as a Representative?

For example, can we represent Shubman's or Yashasvi's batting in this series with one number? Discuss.

We saw we may already — the total number of runs in the group! But, if the groups sizes are different, then the total may not be an appropriate measure to compare.

In some matches, a player could have scored more and in other matches less. A representative number for the group can be found by balancing out all the matches and divide the total by the number of matches played. We call this value the 'average' or 'arithmetic mean'^? of the given data.

Here, the average number of runs scored by a player in a match = (Total runs scored by the player) ÷ (Number of matches played).

Average number of runs scored by Shubman is \(110 \div 5 = 22\) runs.
Average number of runs scored by Yashasvi is \(96 \div 4 = 24\) runs.

In this series, Yashasvi's average number of runs is higher than Shubman's.

Average as Fair-Share

The average can be understood as fair-share or equal-share.

Shreyas and 4 of his friends have collected the following numbers of guavas: 3, 8, 10, 3, and 4. Parag and 5 of his friends have collected the following numbers of guavas: 5, 4, 6, 3, 4, and 8. Each group will share their guavas equally among themselves. In which group will each member get a bigger share of guavas?

To find this out, we first find out how many guavas each group has collected. Then we divide this total by the number of people in the group to get each member's share.

Shreyas's group has collected \(3 + 8 + 10 + 3 + 4 = 28\) guavas. Each member of Shreyas's group gets \(28 \div 5 = 5.6\) guavas.

Parag's group has collected \(5 + 4 + 6 + 3 + 4 + 8 = 30\) guavas. Each member of Parag's group gets \(30 \div 6 = 5\) guavas.

So, the members of Shreyas's group will have more guava each than the members of Parag's group.

Hibiscus flower example: Vanshami tracks the number of Hibiscus flowers blooming in her garden each day. The data for the last five days is: 2, 7, 9, 4, 3. What is the average number of Hibiscus flowers blooming per day in Vaishnavi's garden?

Average = (total number of flowers bloomed) ÷ (number of days) = \((2 + 7 + 9 + 4 + 3) \div 5 = 25 \div 5 = 5\).

On an average, 5 Hibiscus flowers bloom each day. In this evening, 5 flowers bloomed, making 5 × 5 = 25 the total number of flowers bloomed over five days; one day, four flowers bloomed less than this average, but on another day, there will likely be more.

Figure it Out (Representative Values)

Know Your Onions!

The table shows the monthly price of onions, in rupees per kilogram (₹/kg), at two towns. Where are onions costlier, according to you?

Data can be described and compared by referring to its minimum value, maximum value, the total of all its values, and the difference between the maximum and minimum values.

A Visual Approach — The Dot Plot

To study data, we can visualise it in multiple ways. One way is shown below — it is called a dot plot^?. Dot plots show data points as dots on a line, helping us visualise variability and patterns in data.

The prices in Yahapur are in green and those in Wahapur are in orange. The horizontal line shows the prices from 10 to 60 (instead of starting from 0 as there are no values from 0 to 10 or above 60). The dots on the vertical line give the number of occurrences of a data value.

Finding the Average Onion Price

A statement such as, "The price of onions is ₹35 per kilo", may not trigger any further questions. But looking at variations in data, like the prices of onions over a year in Yahapur and Wahapur, can spark one's curiosity. You might wonder — Do the seasons affect the price? Are there factors that determine the price? How much do onion prices vary? How do the price fluctuations impact farmers, consumers, and the industry?

Averages Around Us

The Arithmetic Mean is frequently used in statistics, mathematics, experimental sciences, economics, sociology, sports, biology, and diverse disciplines. Some statements involving averages in different scenarios: