# Statistics – Important Formulas and Concepts for Placement Aptitude

Statistics is not the most commonly tested topic for Placements but it is definitely one of the easy topics. So if you get even few questions from this topic, it will be an advantage for you as you can score some easy marks quickly in quant section

## Measures of Central Tendencies

A measure of central tendency indicates the central value of the size of a typical member of the group

Various measures discussed under central tendency are –

- Mean (also called “Arithmetic Mean” or “Average”)
- Median
- Mode

### Arithmetic Mean or Average (A.M.) \( (\bar{x}) \)

Given x_{1}, x_{2}, x_{3}, ….. x_{n} (n individual items), then – $$AM= \overline{x} = {\LARGE [} \ \frac{x_{1}+x_{2}+x_{3}…+x_{n}}{n} \ {\LARGE ]} \\ \overline{x} = \frac{Sum \ of \ the \ observations}{Number \ of \ observations}$$

The arithmetic mean to two numbers a, b is \( \frac{a+b}{2} \)

**Q: What is the mean of given aptitude score of the students out of 100 in placement tests. If only students with higher than mean will be qualified for the next round, who all cleared it?**

**Arun – 20, Bala – 90, Chandra – 40, Debolina – 50, Priyanka – 60**

Sol: Mean = \( \frac{20+90+40+50+60}{5}=\frac{260}{5}=52 \)

Only Bala and Priyanka cleared the test

### Median

The median is the middle value of a dataset when it is arranged in ascending or descending order. If the dataset has an odd number of values, the median is the middle value.

**Q: What is the median of run scored by a new Indian batsman in 5 matches: 25, 14, 40, 17, 22?**

Sol: First arrange the numbers in increasing order: 14, 17, 22, 25, 40

Hence the median is 22 which is the central number

If the dataset has an even number of values, the median is the average of the two middle values

**Q: Calculate median rainfall (in mm) in Jaisalmer for given 5 monsoon days: 3, 5, 9, 11, 13,15 **

Sol: Median = \ \frac{9+11}{2}=10mm\)

### Mode

Mode is the value which is most often found in the given set of observations, i.e, the value occurring the highest number of times.

**Q: Given are the feet sizes (rounded to nearest standard size) of population sample in India, which size should be manufactured by the new shoe brand first to maximize the sale?**

7, 8, 6, 9, 9, 9, 8, 8, 9, 9, 6, 10, 11, 8, 9

Sol: Note that question doesn’t ask you to use mode here directly. To maximize a sale you need to find what is the most frequent size, which is the mode of the sample.

- 6 occurs 2 times
- 7, 10 and 11 occur once
- 8 occurs 4 times
- 9 occurs 6 times

Mode = 9, which is what company should manufacture first

**Q: Consider the following set of numbers: 5, 12, 7, 5, 9, 5, 12. Find the mean, median, and mode of the data**

Sol: **Mean**: Add all the numbers and divide by the number of numbers.

Mean = (5+12+7+5+9+5+12)/7 = 55/ 7 ≈ 7.86

So, the mean is **7.86**

**Median**: Arrange the numbers in ascending order and find the middle number.

Arranged order: 5, 5, 5, 7, 9, 12, 12

The middle number is the 4th number, which is 7. So, the median is 7

**Mode**: The number that appears most often is 5 (appears 3 times).

So, the mode is 5

**Q:** The ages of a group of 8 students are as follows: 15, 17, 16, 18, 20, 15, 19, and 21. Find the mean, median, and mode of their ages.

Sol: **Mean** = (15+17+16+18+20+15+19+21) / 8 = 141/8 = 17.625

**Median**: First, arrange the ages in ascending order: 15, 15, 16, 17, 18, 19, 20, 21.

Since there are 8 numbers, the median is the average of the 4th and 5th numbers: Median = (17+18)/2 = 17.5

So, the median age is 17.5.

**Mode**: The number that appears most frequently is 15. So, the mode is 15.

You can also refer following videos to grasp basics of the topic in more detail:

Refer Topic: Algebra: https://www.learntheta.com/placement-aptitude-algebra/

Read more about LearnTheta’s AI Practice Platform: https://www.learntheta.com/placement-aptitude/