### Summation

\displaystyle\sum_{i=1}^n

We use Sigma symbol to represent a summation.

Instead of saying:

x_1 + x_2 + x_3 + x_4 + x_5 + x_6

We can write:

\displaystyle\sum_{i=1}^6x_i

## Measures of Spread

**Five Number Summary** – gives values for calculating the range and interquartile range.

**Minimum**– the smallest number in the dataset.**Q**_{1}– The value such that 25% of the data fall below.**Q**– The value such that 50% of the data fall below._{2}**Q**– The value such that 75% of the data fall below._{3}**Maximum**– The largest value in the dataset.

**Range** – calculated as the difference between the maximum and the minimum.

range = maximum - minimum

**IQR (Interquartile Range) **– calculated as the difference between Q_{3} and Q_{1}

IQR = Q_3 - Q_1

**Steps to compute:**

- Arrange data set from least to highest number.
- Get the lowest number as the
**minimum**. - Get the highest number as the
**maximum**. - Get the median/mean as
**Q**._{2} - Get the median/mean of the first data set as the
**Q**. (don’t include the median/mean of the entire data set Q_{1}_{2}.) - Get the median/mean of the second data set as the
**Q**. (don’t include the median/mean of the entire data set Q_{3}_{2}.) - Get the difference of the maximum and minimum as the
**range**. - Get the difference of the Q
_{3}and Q_{1}as the**IQR**.

**Examples**

```
1, 5, 10, 3, 8, 12, 4, 1, 2, 8
1. 1, 1, 2, 3, 4, 5, 8, 8, 10, 12
2. Minimum = 1
3. Maximum = 12
4. Q
```_{2} = 4+5 = 9/2 = **4.5**
5. Q_{1} = **2**
6. Q_{3} = **8**
7. Range = 12-1 = **11**
8. IQR = 8-2 = **6**

```
5, 10, 3, 8, 12, 4, 1, 2, 8
1. 1, 2, 3, 4, 5, 8, 8, 10, 12
2. Minimum = 1
3. Maximum = 12
4. Q
```_{2} = 5
5. Q_{1} = 2+3=5/2 = 2.5
6. Q_{3} = 8+10=18/2 = 9
7. Range = 12-1 = 11
8. IQR = 9-2.5 = 6.5

**Box Plot** – are useful for quickly comparing the spread of two data sets across some key metrics, like quartiles, maximum, and minimum.

- The beginning of the line to the left of the box and the end of the line to the right of the box represent the minimum and maximum values in a dataset.
- The visual distance between these markings is an indication of the range of the values.
- The box itself represents the IQR. The box begins at the Q
_{1}value, ends at the Q_{3}value, and Q_{2}, or the median, is represented by a line within the box.

## Standard Deviation and Variance

**Standard Deviation** – on average, how much each point varies from the mean of the points.

**Variance** – average squared difference of each observation from the mean.

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

**How to Calculate Standard Deviation**

```
Dataset=
10, 14, 10, 6
```

- Calculate the mean.

(\sum_{i=1}^4 x_i)/n \\ 10+14+10+6 \\ 40/n \\ 40/4 \\ =10

- Calculate the distance of each observation from the mean and square the value.

(x_i - \bar{x})^2 = \\ (10-10)^2 = 0^2 = 0 \\ (14-10)^2 = 4^2 = 16 \\ (10-10)^2=0^2=0 \\ (6-10)^2=-4^2=16

- Calculate the
**variance**, the average squared difference of each observation from the mean.

\sqrt {\frac 1 n \displaystyle\sum_{i=1}^n (x_i - \bar{x})^2} \\ (0+16+0+16)/4\\32/4\\=8

**The variance is 8. **

- Calculate the standard deviation, the square root of the variance.

\sqrt 8 \\ =2.83

**The standard deviation is 2.83.**

#### Sample Problem

```
Dataset
1, 5, 10, 3, 8, 12, 4
```

**Mean**– 6.14

(\sum_{i=1}^7 x_i)/7 \\ 1+5+10+3+8+12+4=43\\43/7\\=6.14

- Variance

(x_1-\bar{x})^2\\(1-6.14)^2=-5.14^2=26.42\\(5-6.14)^2=-1.14^2=1.30\\(10-6.14)^2=3.86^2=14.90\\(3-6.14)^2=-3.14^2=9.86\\(8-6.14)^2=1.86^2=3.46\\(12-6.14)^2=5.86^2=34.34\\(4-6.14)^2=-2.14^2=4.58\\94.86/7\\=13.55

- Standard Deviation

\sqrt {13.55} \\ =3.68