If there are two metrics and a large data set for each metric, which methods you can use to detect a relationship between these metrics?

There are several statistical methods that can be used to detect a relationship between two metrics. Some commonly used methods include: 

Pearson's correlation coefficient: This measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 indicates no relationship, and 1 indicates a perfect positive linear relationship.

Spearman's rank correlation coefficient: This is similar to Pearson's coefficient, but it is used when the two variables are not necessarily linearly related. It also ranges from -1 to 1, with the same interpretation as Pearson's coefficient.

Chi-squared test: This is a statistical test that is used to determine whether there is a relationship between two categorical variables. It calculates the difference between the observed frequencies of the variables and the expected frequencies if there was no relationship, and determines whether the difference is statistically significant.

T-test: This is a statistical test that is used to determine whether there is a significant difference between the means of two groups. It can be used to compare two metrics if the data can be divided into two groups based on some criteria.

In general, the appropriate method to use depends on the type of data and the nature of the relationship between the two metrics. It's important to carefully consider the data and the research question at hand before choosing a statistical method.

Difference between weighted average and simple average (Explain with numbers)

 A weighted average is an average that takes into account the relative importance or weight of each value in a data set. A simple average, also known as the arithmetic mean, is an average that treats all values in the data set as having equal importance.

Here is an example to illustrate the difference between the two. Let's say you have a data set with the following values: 2, 4, 6, and 8. The simple average of this data set would be (2+4+6+8)/4 = 5. The weighted average, on the other hand, would be calculated by multiplying each value by its corresponding weight and then dividing by the sum of the weights. For example, if the weights were 0.5, 1, 1.5, and 2, the weighted average would be (20.5+41+61.5+82)/(0.5+1+1.5+2) = 6.

As you can see, the weighted average takes into account the relative importance of each value in the data set, while the simple average does not. In this case, the weighted average is higher than the simple average because the higher values in the data set have more weight.

Which is Best between Mean, Median and Mode?

 It really depends on the data set and what you are trying to do with it. Here is a brief explanation of each of these statistical measures:

Mean: The mean is the average of a set of numbers. It is calculated by adding up all the numbers in a data set and dividing by the number of items in the set.
Median: The median is the middle value in a data set. To find the median, you need to first order the data set from least to greatest, and then find the value that is in the middle of the set. If there is an even number of items in the data set, the median is the average of the two middle numbers.
Mode: The mode is the most common value in a data set. It is the value that appears most often in the data set.

In general, the mean is a good measure of central tendency when the data set is continuous and symmetrical. The median is a good measure when the data set is skewed or has outliers. And the mode is a good measure when the data set is categorical.

Here are some examples of scenarios where one of these measures might be better suited than the others:
If you are analyzing the heights of a group of people, the mean would be a good measure to use because height is a continuous variable and it is unlikely that there are any extreme outliers in the data set.
If you are analyzing the ages of a group of people, the median would be a better measure to use because age is a continuous variable and it is likely that there are some extreme outliers in the data set (e.g. very young or very old individuals).
If you are analyzing the favorite colors of a group of people, the mode would be the best measure to use because the color is a categorical variable and there is no inherent order to the values.

It's important to note that these are just general guidelines, and the best measure to use really depends on the specific data set and what you are trying to do with it. In some cases, it might be appropriate to use multiple measures to get a more complete picture of the data.