Stat= Social the relationship between variables
Variables= any characteristic of the object described in the data that varies
Population- entire group of individuals we want info about
Sample- group we collect data from
-use to draw conclusions about populations
-when we take different samples, we get different values in the solutions. This is called variability in sampling
Ideally, the goal is we want to have small variability. To show that you are making close and
accurate/better samples.
Parameter- # describing population
-it is an unknown fixed number
Statistic- #describing sample
-estimate of unknown parameter
Pop parameter unknown fixed #
Sample statistic estimated # of unknown parameter
Bias- consistent repeated deviation of statistic from parameters in the same deviations
-goal is to have little bias
-also goal to have small variability
-reasons for small variability:
1. Large samples reduce variability (diminishing marginal utility)
2. Samples have to be random (*very key*). Randomness reduces biases. Randomness is
every individual has equal likelihood of being in sample
Bad sampling- not representative of the population
-convenience sampling= get data from individuals that is easy to find/access
-voluntary response sampling= a general appeal that individuals respond to
If a question asked has 2 different topics in the question, that question isnt really valid because you could be for one part of the
question but against the other. These are called double-barreled questions (2 questions in 1)
-the wording of questions has a huge impact on the responses (issues of wording in questions challenges the respondents)
-operationalization= process of measuring concepts and converting it to variable(s)
-when doing research, we take a concept and convert it to a variable to measure the issue
-is it valid way to get at the concept?
-how accurate are the measurements?
-ex: income, social class
-asked your gross income in 2011; if the individual just filed tax returns, then the number
should be pretty accurate; but if a person is making an estimated guess would less accurate
-Validity- measures what it suppose to
-Reliability- measuring something consistently
- for both these, an example of a concept would be overall happiness and you are asking a question how
happy are you today? That question would not be valid because its is just today, not overall happiness
-these can work together, but not always: could have validity and not reliability or vice versa
-is it valid? Is it reliable? Is it consistent?
Who is doing the research and what is being left out of the research?
-ex. Athletes are drinking soda to increase their athletic performance
-you have to really look into the topic and make sure that it is reliable
-ex number of Americans living below the poverty grew from the 1979 to from 26 million to 34 million
You cant say the poverty rate increased because the population could have also grew
Trends
What does the distribution look like?
Distribution of a variable tells us what values it takes and how often it takes those values
Looking for the range of options and how often those options are shown/present
Mean=average
Mean average= the sum of all the values in a group, divide by the number of values in a group down on board
List the entire set of values
Compute the sum of values
Divide the sum by the number of values
The example we did in class what height sizes in our own class
Total sum was 1,514 in
N=23
Mean= 65.83
Median = middle
1. Arrange all values in order of size from smallest to largest
2. Find the middle position
M=n+1/2 if its an odd number of values (counting from the first value
N+1/2 23+1/2 24/2= 12
If its an even number you n/2 and n/2 +1 and take the average
So if we had the teacher in we get: 24/2 =12 and 24/2 + 1 = 13
Median= 66 [get info for this problem]
Mode= most frequently occurring value
List the values in order of size
See which value occurs the most
The mean is majorly affected by outliers (outliers are extreme values)
The median is a better representation to the trends because it is not affected by extreme outliers
So often good data reports, will show both the mean and the median
Sometimes the bell graph will skew to the right/ left if there are more values to the right then the graph will skew to the right
Skew to the right (positive) having large outliers; will drag the distribution to the right; causing the
mean to be larger than the median)
Skew to the left (negatives) having small outliers (very small redwoods); drag the distribution to the
left; causing the mean to be smaller than the median
Measurement of Variance they measure how spread out the data
-how the values are spread out
-what the range is
-the more spread out our data the greater the variance
5 Number Summary= helpful but limited when finding standard deviation
(Mean And Standard deviation)
List values in
Find median
Find the median of median =Q1
Find the median of values below median =Q3
Ex. Minimum = 57
Maximum= 73
Q1=63
Q3= 69
The range= max min 73 57= 16 values
The Inter Quartile Range (IQR)
Q3 Q1
Standard deviation- the average distance of each value in the range from the mean
5 steps for Standard Deviation
Find the mean
Find the distance of each value from the mean
Square the difference
Sum the squared distances
Divide by (n 1)
Variance
Calculating the average squared distance from the mean
Symmetric distribution- technically the graph have should the same image if it was cut directly in half
A type of symmetrical distribution that is not shaped like a bell curve is a bimodal symmetric distribution. This graph is
shaped like two bell curves placed side by side. The two sides of this graph still mirror each other; however, only the mean and median
occur at the same point - the center of the graph. The modes occur at two points: the highest point in each of the two bell curves.
If there are gaps between values then there are outliers
Unimodal, Bimodal, and Multimodal Distributions
Density curve- like drawing line over the tops of a histogram
Shows the proportion of observations in the area under the curve
The total area under the density curve equals 1
Shows center and spread of distribution
Intended to reflect the idealized shape of the population distribution
Used with large ns
The area under the curve can be used to describe the relationships between individual values and the rest of the
distribution
Mean and Median of Density curves
Mean of a density curve is the point which the curve balances
When a density curve is symmetric, the mean and the median are the same
http://www.math.utah.edu/~hughes/Chapter_03.pdf
Normal Distribution:
Symmetric
Bell shaped
Unimodal
No outliers or gaps
-if Normal Distribution, then use mean and standard deviation
When the distribution is positively skewed or skewed to the right
Implies mean is bigger than median
When the distribution is negatively skewed or skewed to the left
Implies median is bigger than mean
Standardizing
Compare individual values to the rest of the distribution
Compare across units
Z-score= standardized values that use information about the mean and the standard deviation
Unit of z-score is standard deviation from the mean
Z-score= (x-x^)/s
Z-score = standard score
Standard score= (observation mean) / standard deviation
z-score shifts the distribution so the mean becomes 0 (zero) and the standard deviation becomes 1
Normative Model
When we are standardizing
Mean= 0
Standard deviation= 1
Finding the z-score on the normal model will show the proportion of individuals who score above or below a certain
point
Graph:
Area under curve =1
Symmetric
68 95 97% Rule
68% of the observations fall within one standard deviation of the mean
95% of the observations fall within 2 standard deviations of the mean
99.7% of the observations fall within 3 standard deviations of the mean
Normal Distributions
Normal distributions are a specific type of distribution that have by bell-shaped, symmetric density curves. There are many naturally
occurring variables that tend to follow normal distributions. Some examples are biological characteristics, IQ tests, and repeated
measurements taken by physcial devices.
A normal distribution can be completely described by its mean and standard deviation.
For all normal distributions, 68% of the data will fall within one standard deviation of the mean, 95% will fall within two standard
deviations of the mean, and 99.7% will fall within three standard deviations of the mean. This is called the 68-95-99.7 rule.
Bi-variate Relationships- how two variables are related/association/connected to each other (avoid the word causal) (height and
weight)
We have to think how the variables affect/impacts each variable (ex. More education impacts more income)(height
impacts/affects weight)
Types of variables:
Independent (explanatory) (does the impacting/affecting) (explains the change in the dependent variable)
Dependent (response) (the variable that is impacted/affected)
Crosstabuluation (crosstabs)
Used for categorical variables
There are columns and rows
The columns are independent variable (if time is one of the variable, it is the independent variable)
The rows are the dependent variables
-HISTOGRAMS: To construct a histogram, all you do is plot the frequencies on the vertical axis and the class intervals on the
horizontal axis.
The goal of a graphical display is to provide a visual impression of the characteristics of the data from a sample. The hope is that the
characteristics of the sample are a likely indication of the characteristics of the population from which it was drawn.
- When we examine histograms (or stem plots), we will b e always b e interested in the overall pattern and any striking deviations
from that pattern; in particular:
- center of the distribution of data
- spread (variation) in the distribution of data
- shape: is the distribution symmetric or skewed? Symmetric distributions are those whose left and right-hand sides lo
ok like mirror images of one another perfect symmetry is a rarity in real life).
- the presence of outliers (i.e., \unusual observations").
-Stem plots provide a display of the distribution while retaining the numerical values themselves. The idea is to separate each data
value into a stem and a leaf .
Stem plots work well with small-to-moderate-sized data sets, say, 15 to 50 observations.
DIRECTIONS: To create a stem plot for this data (see textbook for full instructions).
1. Separate each of the observations into 2 parts: the rightmost digit (the leaf ) and the remaining digits
(the stem). Rounding may b e necessary.
2. Vertically list the levels of the stem in ascending order (smallest on top) and draw a vertical line to its
right.
3. Write each leaf to the right of the appropriate stem. For a given stem, the leaves should b e in
ascending order from left to right.