Chapter 1 Review

Why Statistics?

The Birth of Statistics Began in the 17th Century System to combine probabilities with Bayesian inference Important in the development of: o Psychology o Sociology o Thermodynamics o Gambling o Computer Science

Important Vocabulary

The Science of Statistics Descriptive Statistics o Look for patterns o Summarize information o Present in convenient form Inferential Statistics o Used to make estimates/predictions for FUTURE

The Context of Population

Sample Variance

for n measurements: sum of the squared distances from the mean divided by (n-1)

1. Find the mean 2. Subtract the mean from each distinct n and square the difference 3. Multiply by the frequency 4. Add the resulting numbers 5. Divide by (n-1) Steps for finding Sample Variance:

Why (n-1)? important property of sample statistics that estimates a population parameter: if you evaluate the sample statistics for every possible sample and find the average, the average of the sample statistics should equal the population parameters

dividing by (n-1) in the sample variance s² is unbiased estimator of the population variance σ²

Sample Variance Problem You are considering adding a new stock that has an expected return of 4%. Before you buy this stock, you would like to know its variance. Using historical data, calculate the variance of this stock's returns.

Sample Variance Problem 1. Find the mean xfxf mean= 0.48/12 =0.04

Sample Variance Problem 2. Subtract the mean from each distinct n and square the difference x(x-0.04)(x-mean)²

Sample Variance Problem 3. Multiply by the frequency (x-mean)²f(x-mean)²f

Sample Variance Problem 4. Add the resulting numbers (x-mean)²f sum=0.0026

Sample Variance Problem 5. Divide by (n-1) = = s² (12-1)

a measure of the dispersion in a distribution Sample Standard Deviation

positive square root of the sample variance

1. Find the Sample Variance 2. Take the positive square root Steps for finding Sample Standard Deviation:

Sample Standard Deviation Problem You are considering adding a new stock that has an expected return of 4%. Before you buy this stock, you would like to know its variance. Using historical data, calculate the standard deviation of this stock's returns.

Sample Standard Deviation Problem From the earlier problem, we found that the variance was: = = s² (12-1) sqrt ( ) = s =.01536

ratio of the standard deviation to the mean Coefficient of Variance

Why use CV? another way to describe variation describes dispersion as a percentage of the mean <5% generally mean good performance Formula

CV problem Data from experiments show an SD of 4.00 mL at a concentration of 100. mL and an SD of 8.00 mL at a concentration of 200. mL. What are the CVs? 4.00 = 4.00% 8.00 = 4.00%

Box Plot

- A Box Plot is a diagram which helps with the organization of data. - On a typical Box Plot, you will be able to find the minimum, maximum, lower quartile (Q 1 ), median (Q 2 ), upper quartile (Q 3 ), and the mean. What is a Box Plot?

- The Range is equal to Maximum - Minimum - The Inter-quartile range (IQR) is equal to Upper Quartile - Lower Quartile - In a Box Plot, the mean and the median are always found in between the lower and upper quartiles Organization of the Box Plot Data

- If the mean and median are equal, the graph will be symmetric Mean Vs. Median Mean = Median

- If the mean is greater than the median, the graph will be skewed to the right. Mean Vs. Median Mean > Median

- If the mean is less than the median, the graph will be skewed to the left Mean Vs. Median Mean < Median

For the first quiz in an Algebra class, students received the following grades - Draw a six-point box plot for the data - Determine if the data is either skewed to the right, left, or neither. Box Plot Problem x f

- The minimum would be the lowest score. The minimum would be 0 - The maximum value would be the highest score. This would be 10 - To find the mean, you have to multiply the frequencies with the x- values and divide them by the number of frequencies. The mean would be 5 Find the Minimum, Maximum, Mean xfxf Totals Mean = xf / f = 255 / 51 = 5

Find the Lower Quartile (Q 1 ) - To find the Lower Quartile, you have to find the 25 th percentile. - You can do this by multiplying 25% with the number of frequencies (51) P 25 = 25 / 100 x 51 = > 13 Q 1 should be the 13 th term on the chart. - P 25 is the 13 th term which has a value of 3, which makes Q 1 = 3 xfxf Totals th term ->

Find the Median (Q 2 ) - To find the Median, you have to find the 50 th percentile. - You can do this by multiplying 50% with the number of frequencies (51) P 50 = 50 / 100 x 51 = > 26 Q 2 should be the 26 th term on the chart. - P 50 is the 26 th term which has a value of 5, which makes Q 2 = 5 xfxf Totals th term ->

Find the Upper Quartile (Q 3 ) - To find the Upper Quartile, you have to find the 75 th percentile. - You can do this by multiplying 75% with the number of frequencies (51) P 75 = 75 / 100 x 51 = > 39 Q 3 should be the 39 th term on the chart. - P 75 is the 39 th term which has a value of 7, which makes Q 3 = 7 xfxf Totals th term ->

Drawing the Box Plot - Now that we have the data, draw the box plot. - This is what the Box Plot would look like with the data of Minimum(0), Lower Quartile(3), Mean(5), Median(5), Upper Quartile(7), Maximum(10): Minimum 0 Maximum 10 Lower Quartile 3 Upper Quartile 7 Median 5 Mean 5 - Since the mean and the median are equal, the data is neither skewed to the left nor right. It is Symmetric

Box Plot Using a Calculator Faster and easier way to calculate the quartiles using a Ti-83 or Ti-84 calculator Reduces risk of arithmetic error Reduces time creating a box plot Increases overall accuracy of calculation

Steps on How Create a Box Plot Press [STAT] Select [Edit] **Clear old data before inputting new** Fill in [L1] column (This will be your x values) Then fill in your [L2] column (This will be your frequency values)

Calculating Your Values Press [STAT] again Select [Calc] Tab Select [1-Var Stats] **Make sure your L1 & L2 lists are being used in the proper section** Then press [ENTER] 3 times to load your calculation x(Days of the week) f(Frequency of college kids drinking) Monday14 Tuesday26 Wednesd ay 310 Thursday413 Friday525 Saturday636 Sunday720

Your values should be as listed Min = 1 Q 1 = 4 Median = 5 Q 3 = 6 Max = 7 Mean = 5.08 Minimum 1 Lower Quartile 4 Median 5 Mean 5.08 Upper Quartile 6 Maximum 7