LET’S RECAP ALL THE DESCRIPTIVE STATISTICS THAT I POSTED BEFORE,

**5 number summary**

The five-number summary is a descriptive statistic that provides a summary of a dataset. It consists of five values that divide the dataset into four equal parts, also

known as quartiles. The five-number summary includes the following values:

- Minimum value: The smallest value in the dataset.
- First quartile (Q1): The value that separates the lowest 25% of the data from the rest of the dataset.
- Median (Q2): The value that separates the lowest 50% from the highest 50% of the data.
- Third quartile (Q3): The value that separates the lowest 75% of the data from the highest 25% of the data.
- Maximum value: The largest value in the dataset.

The five-number summary is often represented visually using a box plot, which displays the range of the dataset, the median, and the quartiles.

The five-number summary is a useful way to quickly summarize the central tendency, variability, and distribution of a dataset.

**Interquartile Range (IQR)**

The interquartile range (IQR) is a measure of variability that is based on the five-number summary of a dataset. Specifically, the IQR is defined as the difference between the third quartile (Q3) and the first quartile (Q1) of a dataset.

**Boxplots**

1. What is a boxplot

A box plot, also known as a box-and-whisker plot, is a graphical representation of a dataset that shows the distribution of the data. The box plot displays a summary of the

data, including the minimum and maximum values, the first quartile (Q1), the median (Q2), and the third quartile (Q3).

**How to Create Boxplot with examples**

**Quartiles Formula**

Suppose, Q_{3} is the upper quartile is the median of the upper half of the data set. Whereas, Q_{1} is the lower quartile and median of the lower half of the data set. Q_{2} is the median. Consider, we have n number of items in a data set. Then the quartiles are given by;

Q_{1} = [(n+1)/4]th item

Q_{2} = [(n+1)/2]th item

Q_{3} = [3(n+1)/4]th item

Q1 = 2.75

NOTE: 2.75 is not an integer, so Q1 iwill be

6, 213, 241, 260, 281, 290, 314, 321, 350, 1500**Q1 =**213+0.75(241-213) i .e [2.75 means take 2 and 3 position, and remaining 0.75 will multiple to (subtraction of 3 to 2 position)]

** Benefits of a Boxplot**

○ Easy way to see the distribution of data

○ Tells about the skewness of data

○ Can identify outliers

○ Compare 2 categories of data

**Scatterplot**

```
To create a scatter plot between price and area, you can use a data visualization library like Matplotlib in Python. Here's a simple example:
import matplotlib.pyplot as plt
import numpy as np
# Sample data (replace this with your own dataset)
price = np.array([200000, 300000, 250000, 400000, 350000, 500000])
area = np.array([1500, 2000, 1800, 2500, 2200, 3000])
# Create a scatter plot
plt.scatter(area, price, color='blue', marker='o')
# Add title and labels
plt.title('Scatter Plot: Price vs. Area')
plt.xlabel('Area (sq. ft)')
plt.ylabel('Price ($)')
plt.grid(True)
# Show the plot
plt.show()
```
In this example:
- `price` represents the price of properties.
- `area` represents the corresponding area of the properties.
You can replace the sample data with your own dataset. The `scatter` function from Matplotlib is used to create the scatter plot. Customize the color, marker, and other plot parameters according to your preferences.
If you're working with a dataset in a file (e.g., CSV), you can use libraries like Pandas to read the data and then create the scatter plot. For instance:
```python
import matplotlib.pyplot as plt
import pandas as pd
# Read data from a CSV file (replace 'your_data.csv' with your actual file name)
data = pd.read_csv('your_data.csv')
# Extract 'price' and 'area' columns from the DataFrame
price = data['price']
area = data['area']
# Create a scatter plot
plt.scatter(area, price, color='blue', marker='o')
# Add title and labels
plt.title('Scatter Plot: Price vs. Area')
plt.xlabel('Area (sq. ft)')
plt.ylabel('Price ($)')
plt.grid(True)
# Show the plot
plt.show()
```

**Covariance**

When dealing with multiple variables, calculating the variance of each variable individually may not capture the entire picture of their relationships. The problem arises when we want to understand how two variables vary together or whether changes in one variable are associated with changes in another

•** What is covariance and how is it interpreted?**

Covariance is a statistical measure that describes the degree to which two variables are linearly related. It measures how much two variables change together, such that when

one variable increases, does the other variable also increase, or does it decrease? If the covariance between two variables is positive, it means that the variables tend to move together in the same direction. If the covariance is negative, it means that the variables tend to move in opposite directions. A covariance of zero indicates that the variables are not linearly related.

•** Disadvantages of using Covariance**

One limitation of covariance is that it does not tell us about the strength of the relationship between two variables, since the magnitude of covariance is affected by the scale of the variables.

**Correlation**

2. What is correlation?

Correlation refers to a statistical relationship between two or more variables. Specifically, it measures the degree to which two variables are related and

how they tend to change together. Correlation is often measured using a statistical tool called the correlation coefficient, which ranges from -1 to 1. A correlation coefficient of -1 indicates a perfect negative correlation, a correlation coefficient of 0 indicates no correlation and a correlation coefficient of 1 indicates a perfect positive correlation.

**Correlation and Causation**

The phrase “correlation does not imply causation” means that just because two variables are associated with each other, it does not necessarily mean that one causes the other. In other words, a correlation between two variables does not necessarily imply that one variable is the reason for the other variable’s behaviour.

Suppose there is a positive correlation between the number of firefighters present at a fire and the amount of damage caused by the fire. One might be tempted to conclude that the presence of firefighters causes more damage.

However, this correlation could be explained by a third variable – the severity of the fire. More severe fires might require more firefighters to be present and also cause more damage.

Thus, while correlations can provide valuable insights into how different variables are related, they cannot be used to establish causality. Establishing causality often requires additional evidence such as experiments, randomized controlled trials, or well-designed observational studies.

**Covariance and Correlation**

```
import seaborn as sns
import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
df = pd.DataFrame()
from re import X
x = pd.Series([12,25,68,42,113])
y = pd.Series([11,29,58,121,100])
df['x'] = x
df['y'] = y
df
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 3))
# Plot scatterplots on each axes
ax1.scatter(df['x'], df['y'])
ax2.scatter(df['x']*2, df['y']*2)
ax1.set_title("Covariance - " + str(np.cov(df['x'],df['y'])[0,1]))
ax2.set_title("Covariance - " + str(np.cov(df['x']*2,df['y']*2)[0,1]))
print(np.cov(df['x'],df['y'])[0,1])
print(np.cov(df['x']*2,df['y']*2)[0,1])
fig, ax = plt.subplots(1, 3, figsize=(15, 3))
# Plot scatterplots on each axes
ax[0].scatter(df['x'], df['x'])
ax[1].scatter(df['x'], df['y'])
ax[2].scatter(df['x']*2, df['y']*2)
ax[0].set_title("Covariance - " + str(np.cov(df['x'],df['x'])[0,1]))
ax[1].set_title("Covariance - " + str(np.cov(df['x'],df['y'])[0,1]))
ax[2].set_title("Covariance - " + str(np.cov(df['x']*2,df['y']*2)[0,1]))
### Correlation
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 3))
# Plot scatterplots on each axes
ax1.scatter(df['x'], df['x'])
ax2.scatter(df['x'], df['y'])
ax1.set_title("Correlation - " + str(df['x'].corr(df['x'])))
ax2.set_title("Correlation - " + str((df['x']).corr(df['y'])))
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 3))
# Plot scatterplots on each axes
ax1.scatter(df['x'], df['y'])
ax2.scatter(df['x']*2, df['y'])
ax1.set_title("Correlation - " + str(df['x'].corr(df['y'])))
ax2.set_title("Correlation - " + str((df['x']*2).corr(df['y']*2)))
```

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