Calculating Measures of Spread Using Python

dists = msleep['sleep_total'] -
        np.mean(msleep['sleep_total'])
print(dists)

0 1.666265
1 6.566265
2 3.966265
3 4.466265
4 -6.433735
...

sq_dists = dists ** 2
print(sq_dists)

0   2.776439
1   43.115837
2   15.731259
3  19.947524
4  41.392945
...

sum_sq_dists = np.sum(sq_dists)
print(sum_sq_dists))

1624.065542

variance = sum_sq_dists / (83 - 1)
print(variance)

19.805677

np.var(msleep['sleep_total'], ddof=1)

19.805677

np.var(msleep['sleep_total'])

19.567055

np.sqrt(np.var(msleep['sleep_total'], ddof=1))

4.450357

np.std(msleep['sleep_total'], ddof=1)

4.450357
```

`
## Mean absolute deviation
Mean absolute deviation takes the absolute value of the distances to the mean and then takes the mean of those differences. While this is similar to standard deviation, it's not exactly the same. Standard deviation squares distances, so longer distances are penalized more than shorter ones, while mean absolute deviation penalizes each distance equally. One isn't better than the other, but SD is more common than MAD.
```Python
dists = msleep['sleep_total'] - mean(msleep$sleep_total)
np.mean(np.abs(dists))
```
```
3.566701
```
## Quantiles
Before we discuss the next measure of spread, let's quickly go through quantiles. Quantiles, also called percentiles, split up the data into some number of equal parts. Here, we call **np.quantile**, passing in the column of interest, followed by 0.5. This gives us 10.1 hours, so 50% of mammals in the dataset sleep less than 10.1 hours a day, and the other 50% sleep more than 10.1 hours, so this is exactly the same as the median. We can also pass in a list of numbers to get multiple quantiles at once. Here, we split the data into 4 equal parts. These are also called **quartiles**. This means that 25% of the data is between 1.9 and 7.85, another 25% is between 7.85 and 10.10, and so on.

```Python
np.quantile(msleep['sleep_total'], 0.5)
```
```
10.1
```
### Boxplots use quartiles
The boxes in box plots represent quartiles. The bottom of the box is the first quartile, and the top of the box is the third quartile. The middle line is the second quartile or the median.

```Python
import matplotlib.pyplot as plt
plt.boxplot(msleep['sleep_total'])
plt.show()
```

![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/a2frturmku7pdmmo2eiw.png)

### Quantiles using np.linspace()
Here, we split the data into five equal pieces, but we can also use **np.linspace** as a shortcut, which takes in the starting number, the stopping number, and the number intervals. We can compute the same quantiles using **np.linspace** starting at zero, stopping at one, splitting into 5 different intervals.
```Python
np.quantile(msleep['sleep_total'], [0, 0.2, 0.4, 0.6, 0.8, 1])
```
```
array([ 1.9 , 6.24, 9.48, 11.14, 14.4 , 19.9 ])
```
```Python
np.linspace(start, stop, num)
np.quantile(msleep['sleep_total'], np.linspace(0, 1, 5))
```
```
array([ 1.9 , 7.85, 10.1 , 13.75, 19.9 ])
```
## Interquartile range (IQR)
The interquartile range, or IQR, is another measure of spread. It's the distance between the 25th and 75th percentile, which is also the height of the box in a boxplot. We can calculate it using the quantile function, or using the **IQR** function from **scipy.stats** to get 5.9 hours.
```Python
np.quantile(msleep['sleep_total'], 0.75) - np.quantile(msleep['sleep_total'], 0.25)
```
```
5.9
```
```Python
from scipy.stats import iqr
iqr(msleep['sleep_total'])
```
```
5.9
```

## Outliers
Outliers are data points that are substantially different from the others.
But how do we know what a substantial difference is? A rule that's often used is that any data point less than the first quartile - 1.5 times the IQR is an outlier, as well as any point greater than the third quartile + 1.5 times the IQR.

- data < Q1 - 1.5 x IQR or

- data > Q3 + 1.5 x IQR 



### Finding Outliers
To find outliers, we'll start by calculating the IQR of the mammals' body weights. We can then calculate the lower and upper thresholds following the formulas from the previous slide. We can now subset the DataFrame to find mammals whose body weight is below or above the thresholds. There are eleven body weight outliers in this dataset, including the cow and the Asian elephant.
```Python
from scipy.stats import iqr
iqr = iqr(msleep['bodywt'])
lower_threshold = np.quantile(msleep['bodywt'], 0.25) - 1.5 * iqr
upper_threshold = np.quantile(msleep['bodywt'], 0.75) + 1.5 * iqr
```
```
msleep[(msleep['bodywt'] < lower_threshold) | (msleep['bodywt'] > upper_threshold)]
```
```
        name                   vore     sleep_total       bodywt
4      Cow                     herbi         4.0           600.000
20     Asian elephant          herbi         3.9           2547.000
22     Horse                   herbi         2.9           521.00
```

## Using the .describe() method
Many of the summary statistics we've covered so far can all be calculated in just one line of code using the **.describe** method, so it's convenient to use when you want to get a general sense of your data.
```Python
msleep['bodywt'].describe()
```
```Python
count 83.000000
mean 166.136349
std 786.839732
min 0.005000
25% 0.174000
50% 1.670000
75% 41.750000
max 6654.000000
Name: bodywt, dtype: float64
```