Soc 110

What is Statistics

• science of collecting, organizing, and interpreting data

• The Data (numbers/other pieces of information) that describe or summarize something

Parameter vs. Statistic

numerical summary of the population

• EX: ex - % of all American adults who approve of president bidens performance as president

Statistics

• ex. % of 1000 adult Americans in a poll who approve of president bidens performance as president

Data

information that is collected to gain knowledge about a field of to answer a question of interest

Population

total set of subjects of interest

Sample

Subset of the population on which the study collects data on

Variable

Characteristics that can vary across subjects

• Types of variables

  • Quantitative

  • Categorical

  • Discrete

  • Continuous

Quantitative Variables

Numerical values

• EX

  • income, age, years of education

Categorical Variables

Categories

• ex

  • marital status, province of residence, gender, state

Discrete Variable

Separated values

• ex

  • number of siblings

Continuous Variable

Infinite possible values

• ex.

  • Height,

Levels of Measurement

• Nominal

• Ordinal

• Interval

• Ratio

Nominal Level of Measurement

• data that consists of names, labels, or categories only. The data are Qualitative and cannot be ranked or ordered

Examples of the Nominal level of Measurement

Race, Gender, religious affiliation, political party affiliation

Ordinal Level of Measurement

Data that can be arranged in some order (such as low to high). generally does not make sense to do computations with data.

Examples of the Ordinal Level of Measurement

Yelp star rating, low/High/Medium, satisfied/not satisfied, likely/not likely/ (scale surveys).

Interval Level of Measurement

 data in which intervals/distance between measurements are meaningful, but ratios are not. Data at this level have an arbitrary zero point

Examples of the Interval level of Measurement

Temperature in Celsius/ Fahrenheit, IQ/SAT scores, Time/dates,

Ratio Level of Measurement

data in which both intervals and ratios are meaningful. Data at this level have an arbitrary zero point

Examples of Ratio Level of Measurement

Distance, Height, Age, Temperature in Kelvin, weight, speed, Duration(seconds, minutes, etc.)

What are the Measures of Central Tendency

• Mean

• Median

• Mode

Mean

• The Average Value

• Found by —> Sum of all values/total number of all values

• Requires interval-ratio level data

• Sensitive to outliers

Median

Middle Value halfway between the two middle values if the numbers of values are even. If it is odd —> arrange data in ascending order, add two middle numbers - then divide by 2

• ordinal+ level data

• not strongly influenced by outliers

Mode

Most common value(s)

• requires nominal+

What Measures of central tendency can be Nominal

Only Mode

What Measures of central tendency can be Ordinal

Mode, Median, Range

What Measures of Central tendency and variability can be Interval-ratio

Mode, Median, Mean, Range, Standard Deviation

What do Z Scored Measure

 measures how many standard deviations a data point is above or below the population mean

Z Score Formula

Finding Z Score Example:

Middle school principle has 100 8th graders taking a national standardized test with a mean of 400 and a standard deviation of 70. Assuming that the scores are normally distributed. What is the likelihood that one of the graders selected at random will score below 375 on the exam? 

Z= (x - mean)/ standard deviation[little o/sigma] 

Z = (375 - 400)/7

Normal Curve and Z Scores

Percent of Data Set within 1 standard Deviation

68%

Percent of Data Set within 2 standard Deviation

95%

Percent of Data Sent within 3 standard Deviation

99.7%

Three Different Sampling Distributions

• Population Distribution

• Sampling Distribution

• Sample Distribution

Population Distribution

Real distribution representing the characteristics of all members of our population of interest

Descriptive Statistics

Summarize data/ information in a using graphical and statistical representations

• helpful to reduce complexity and present data in a comprehensible manner

Inferential Statistics

provide predictions about a population, based on data from a sample of that populationf

Sampling Distributions

Theoretical probability distribution representing the results of all possible samples drawn from the population 

• used to see how close a sample is to the population mean

Sample Distributions

real distribution describing the characteristics of a sample (Sample subset of the population)

The Larger the sample size?

The smaller the standard error and the closer the sample means would be clustered around the true population parameter

• Principle in Central Limit Theorem

What does Central Limit Theorem use normal curves for?

When the distribution of sample means approximates a normal distribution (nromal bell curve) as the sample size gets larger

• For random sampling with a large sample size n, the sampling distribution of the sample mean y¯ is approximately a normal distribution.

Central Limit Theorem

For random sampling with a large sample size n, the Sampling distribution of the sample mean y¯ is approximately a normal distribution

Inferential statistics

Two Kinds:

• Point Estimate

• Confidence Interval

Point Estimate

When sample statistic used to estimate the exact value of the population parameter (mean, proportion, etc.)

• one of the inferential statistics estimates

Confidence Interval

uses sample statistics to build an interval around which the parameter is likely to fall 

What happens to your range as you become more confident?

It gets larger

• because less room for error

What Confidence Interval do we use in class?

95%

• 1.96

Estimating Population Mean

a parameter of the true average of an entire group, found by claculating sum of all values (N) and dividing by total population size.

• sample mean is best estimate if the population mean when we have only a single sample

Margin of Error

Statistic expressing maximum expected difference between results and true population value

• used to calculate confidence level

Formula for Margin of Error

S - standard Deviation

N - Sample Size

Confidence Level Formula

X(bar) - Sample Mean

E - Margin of Error

Find Margin of Error at 95% and the Confidence Interval

N= 267

Sample Mean of X(bar) = 77.0

SD = 58.6

Margin of Error = 7.03

Confidence Interval = 69.97 to 84.03

What are the Three types of Significance Testing

1. Once Sample T-test

2. Two Sample T-test

3. ANOVA

Once Sample T-test

Compares random sample from a subpopulation against a large population

Two Sample t-test

Compares difference between two sample statistics/two populations

EX - Weekly hours worked for men and for women (Mutually exclusive two samples)

ANOVA

Like a two-sample test, want to know if observed differences in sample means represent:

• uses sampling distribution of variation (differences) in mean

  • EX. levels of support for capital punishment between different age groups ( 20-30, 31-40, 41-50) - different age groups

Sample Significance Testing

Null Hypothesis

• H₀ always States that there is no significant difference between groups

Alternative Hypothesis

• H₁ states that the observed difference really exists in the overall population

What are the Testing Requirments

• Sample was selected randomly via one of the  methods for attaining probability samples 

• The level of measurement is interval scale 

• The sampling distribution is normal in shape 

• Golden standard for testing but can get away without someone

In sampling Significance testing, what happens when p = 0.05?

A p value that is less than or equal to 0.05 is significant. When the p value is significant then we must reject the null hypothesis (H₀ )

P-Value

If the H0 is true, this is the probability that we would see this result

Low P-value means?

• it is unlikely that H0 is true given the result

• equals low probability of being wrong if we reject the null

what goes having a 95% CI mean?