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In this lesson we will explore the relationship between quantitative and categorical variables. We'll use different statistical methods such as mean median differences, side-by-side box plots, and overlapping histograms to try to assess whether or not an association exists between the variables. 

Associations: Quantitative and Categorical Variables

Another way to explore the relationship between a quantitative and categorical variable in more detail is by inspecting overlapping histograms. In the code below, setting `alpha = .5` ensures that the histograms are see-through enough that we can see both of them at once. We have also used `normed=True` make sure that the y-axis is a density rather than a frequency (note: the newest version of matplotlib renamed this parameter `density` instead of `normed`):

```python
plt.hist(scores_GP , color="blue", label="GP", normed=True, alpha=0.5)
plt.hist(scores_MS , color="red", label="MS", normed=True, alpha=0.5)
plt.legend()
plt.show()
```

![title](https://static-assets.codecademy.com/Courses/Hypothesis-Testing/overlapping_dens.svg)

By inspecting this histogram, we can clearly see that the entire distribution of scores at GP (not just the mean or median) appears slightly shifted to the right (higher) compared to the scores at MS. However, there is also still a lot of overlap between the scores, suggesting that the association is relatively weak.

Note that there are only 46 students at MS, but there are 349 students at GP. If we hadn't used `normed = True`, our histogram would have looked like this, making it impossible to compare the distributions fairly:

![title](https://static-assets.codecademy.com/Courses/Hypothesis-Testing/overlapping_freq.svg)

While overlapping histograms and side by side boxplots can convey similar information, histograms give us more detail and can be useful in spotting patterns that were not visible in a box plot (eg., a bimodal distribution). For example, the following set of box plots and overlapping histograms illustrate the same hypothetical data:

![title](https://static-assets.codecademy.com/Courses/Hypothesis-Testing/fake_school_boxplots_and_hist.svg)

While the box plots and means/medians appear similar, the overlapping histograms illuminate the differences between these two distributions of scores.

Inspecting Overlapping Histograms

The difference in mean math scores for students at GP and MS was 0.64. How do we know whether this difference is considered small or large? To answer this question, we need to know something about the spread of the data.

One way to get a better sense of spread is by looking at a visual representation of the data. Side-by-side box plots are useful in visualizing mean and median differences because they allow us to visually estimate the variation in the data. This can help us determine if mean or median differences are “large” or “small”.

Let's take a look at side by side boxplots of math scores at each school:

```python
sns.boxplot(data = df, x = 'school', y = 'G3')
plt.show()
```

![title](https://static-assets.codecademy.com/Courses/Hypothesis-Testing/school_boxplots.svg)

Looking at the plot, we can clearly see that there is a lot of overlap between the boxes (i.e. the middle 50% of the data). Therefore, we can be more confident that there is not much difference between the math scores of the two groups. 

In contrast, suppose we saw the following plot:

![title](https://static-assets.codecademy.com/Courses/Hypothesis-Testing/fake_school_boxplots.svg)

In this version, the boxes barely overlap, demonstrating that the middle 50% of scores are different for the two schools. This would be evidence of a stronger association between school and math score.

Side-by-Side Box Plots

In each of the previous exercises, we assessed whether there was an association between a quantitative variable (math scores) and a BINARY categorical variable (school). The categorical variable is considered binary because there are only two available options, either MS or GP. However, sometimes we are interested in an association between a quantitative variable and non-binary categorical variable. Non-binary categorical variables have more than two categories.

When looking at an association between a quantitative variable and a non-binary categorical variable, we must examine all pair-wise differences. For example, suppose we want to know whether or not an association exists between math scores (`G3`) and (`Mjob`), a categorical variable representing the mother’s job. This variable has five possible categories: `at_home`, `health`, `services`, `teacher`, or `other`. There are actually 10 different comparisons that we can make. For example, we can compare scores for students whose mothers work `at_home` or in `health`; `at_home` or `other`; `at home` or `services; etc.. The easiest way to quickly visualize these comparisons is with side-by-side box plots:

```python
sns.boxplot(data = df, x = 'Mjob', y = 'G3')
plt.show()
```

![title](https://static-assets.codecademy.com/Courses/Hypothesis-Testing/mjob_box_plots.svg)

Visually, we need to compare each box to every other box. While most of these boxes overlap with each other, there are some pairs for which there are some apparent differences. For example, scores appear to be higher among students with mothers working in health than among students with mothers working at home or in an "other" job. If there are ANY pairwise differences, we can say that the variables are associated; however, it is more useful to specifically report which groups are different. 


Exploring Non-Binary Categorical Variables

Recall that in the last exercise, we began investigating whether or not there is an association between math scores and the school a student attends. We can begin quantifying this association by using two common summary statistics, mean and median differences. 
To calculate the difference in mean G3 scores for the two schools, we can start by finding the mean math score for students at each school. We can then find the difference between them:

```python
mean_GP = np.mean(scores_GP)
mean_MS = np.mean(scores_MS)
print(mean_GP) #output: 10.49
print(mean_MS) #output: 9.85
print(mean_GP - mean_MS) #Output: 0.64
```
We see that the mean math score for students at GP is 10.49, while the mean score for students at MS is 9.85. The mean difference is 0.64.
We can follow a similar process to calculate a median difference:
```python
median_GP = np.median(scores_GP)
median_MS = np.median(scores_MS)
print(median_GP) #Output: 11.0
print(median_MS) #Output: 10.0
print(median_GP-median_MS) #Output: 1.0
```
GP students also have a higher median score, by one point.
Highly associated variables tend to have a large mean or median difference. Since “large” could have different meanings depending on the variable, we will go into more detail in the next exercise. 


Mean and Median Differences

In this lesson, we used summary statistics and data visualization tools to examine an association between a quantitative and categorical variable. More specifically, we:

- evaluated mean and median differences
- inspected side-by-side box plots
- examined overlapping histograms
- looked at pair-wise comparisons for a quantitative and a non-binary categorical variable

After calculating a mean or median difference and visually comparing distributions, the next step might be to run a hypothesis test to look for evidence of population-level differences (will a similar difference in scores be observed for ALL students who ever attend these schools?). Now that you know how to investigate whether variables are associated, you can use these techniques to explore associations on more datasets.

Note that data in this lesson was downloaded from the [UCI Machine Learning repository](https://archive.ics.uci.edu/ml/datasets/Student+Performance):

Dua, D. and Graff, C. (2019). UCI Machine Learning Repository [archive.ics.uci.edu/ml/index.php]. Irvine, CA: University of California, School of Information and Computer Science.

The data was originally collected by:

P. Cortez and A. Silva. Using Data Mining to Predict Secondary School Student Performance. In A. Brito and J. Teixeira Eds., Proceedings of 5th FUture BUsiness TEChnology Conference (FUBUTEC 2008) pp. 5-12, Porto, Portugal, April, 2008, EUROSIS, ISBN 978-9077381-39-7.



Review

Examining the relationship between variables can give us key insight into our data. In this lesson, we will cover ways of assessing the association between a quantitative variable and a categorical variable.  

In the next few exercises, we'll explore a dataset that contains the following information about students at two portuguese schools: 
- `school`: the school each student attends, Gabriel Periera (`'GP'`) or Mousinho da Silveria (`'MS'`)
- `address`: the location of the student's home (`'U'` for urban and `'R'` for rural)
- `absences`: the number of times the student was absent during the school year
- `Mjob`: the student's mother's job industry
- `Fjob`: the student's father's job industry
- `G3`: the student's score on a math assessment, ranging from 0 to 20

Suppose we want to know: Is a student's score (`G3`) associated with their school (`school`)? If so, then knowing what school a student attends gives us information about what their score is likely to be. For example, maybe students at one of the schools consistently score higher than students at the other school.

To start answering this question, it is useful to save scores from each school in two separate lists:

```python
scores_GP = students.G3[students.school == 'GP']
scores_MS = students.G3[students.school == 'MS']
```
