Informatics Practices: Class 11 — Chapter 5: Understanding Data

Data that is organized and recorded in a well-defined format is called structured data . It is typically stored in a tabular format consisting of rows and columns, where each column represents an attribute (parameter/characteristic) and each row represents an individual observation.

Table 5.1: Structured data about kitchen items in a shop

Table 5.2: Attributes maintained for different activities

Data that does not conform to a traditional row-and-column structure is called unstructured data . Examples include newspapers (variable layouts, mixed \text and images), emails, web pages, audio/video files, and social media posts.

The Mean (or average) is the sum of all numeric values in an attribute divided by the total number of values.

Mathematical Definition:

Given $n$ values $x_1, x_2, x_3, \dots, x_n$, the mean $\bar{x}$ is computed as:

$$\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$$

Example 5.1:

Height (in cm) of 9 students: [90, 102, 110, 115, 85, 90, 100, 110, 110]

$$\bar{x} = \frac{90 + 102 + 110 + 115 + 85 + 90 + 100 + 110 + 110}{9} = \frac{912}{9} = 101.33\text{ cm}$$

Outlier Warning: An outlier is an exceptionally large or small value compared to the rest of the dataset. The mean is highly sensitive to outliers, which can distort the average. Outliers should be identified and handled before calculating the mean.

The Median is the middle value of a dataset when the values are sorted in ascending or descending order. It divides the dataset into two equal halves.

• If $n$ is odd , the median is the value at the middle position: $\frac{n+1}{2}$.
• If $n$ is even , the median is the average of the two middle values.

Example 5.2:

Sorting the student heights from Example 5.1: [85, 90, 90, 100, 102, 110, 110, 110, 115]

Since $n = 9$ (odd), the median is the value at position 5, which is 102 cm .

The Mode is the value that appears most frequently in a dataset. It can be calculated for both numeric and non-numeric data.

• A dataset has no mode if all values occur only once.
• A dataset can have multiple modes (multimodal) if multiple values share the highest frequency.

Example 5.3:

In the student heights dataset, the value 110 occurs 3 \times (more than any other value). Thus, the mode is 110 cm .

The Range is the difference between the maximum and minimum values in a dataset. It is highly sensitive to outliers.

$$\text{Range} = \text{Maximum} - \text{Minimum}$$

Example 5.4:

For the student heights dataset, the maximum height is 115 cm and the minimum is 85 cm.

$$\text{Range} = 115 - 85 = 30\text{ cm}$$

The Standard Deviation ($\sigma$) measures the spread of data by calculating the positive square root of the average of the squared differences of each value from the mean. Unlike range, it considers all data points.

Mathematical Formula:

$$\sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}}$$

Example 5.5 & Table 5.3: Standard deviation calculation for 9 students ($\bar{x} = 101.33$)

Calculation Steps:

1. Calculate the variance: $$\text{Variance} = \frac{\sum(x - \bar{x})^2}{n} = \frac{938.00}{9} = 104.22$$
2. Calculate the standard deviation: $$\sigma = \sqrt{104.22} \approx 10.2\text{ cm}$$