Chapter 1 – Sampling and Data

Key Terms and Definitions

• Statistics is the science studying the collection, analysis, interpretation, and presentation of data

• Descriptive statistics is the practice of organizing and summarizing data

• Inferential statistics is the use of probability to determine how confident conclusions are

• Populations refer to the an entire group

• Samples are subsets of the population being studied

• Statistics are numerical representations of a property of a sample

• Parameters are numerical characteristics of a population that can be estimated with a statistic

• Representative samples contain the same characteristics as the population

• Variables are characteristics or measurements that can be determined for each member of a population

  • Numerical variables take on values with units
  • Categorical variables place the member into a category

• Data are the variable’s actual values

  • Datum is a single value

• Average is often interchangeably used to refer to the arithmetic mean

Data, Sampling, and Variation

• Qualitative/categorical data describe categories or attributes of a population (e.g. blood type)

• Quantitative data are measured in numbers as the result of counting or measuring attributes (e.g. height)

  • Quantitative discrete data are measured in whole numbers (number of legs on an insect)
  • Quantitative continuous data can have fractions, decimals, etc. (weight of a person)

Visualizing Qualitative Data

• Pie charts use proportional wedges to represent categories
• Bar graphs use vertical or horizontal bars
• Pareto charts use bars descending in categorical size

Percentages can add to >100% if members of the population fall into more than one category.

Sampling Methods

• Simple random sample: n individuals are chosen where each individuals has the same odds of being chosen (drawing numbers from a hat)

• Stratified sample: population is divided into homogenous strata (groups) and a proportionate number is taken from each stratum by simple random sample

  • Cluster sampling divides population into heterogenous clusters

• Systematic sample: a starting point is selected at random and every n^th piece of data is taken

• Sampling with replacement: the chosen member is added back to the population before the next member is chosen

  • Replacement guarantees that samples are independent of each other
  • 10% rule: $n<10\%$ of the population if sampling without replacement

• Convenience sampling: non-random sampling where the most available data is chosen (online surveys)

• Sampling errors are caused by the sampling process

• Non-sampling errors are caused by factors not related to the sampling process

• Sampling biasses are created when the sampling is when some members are more likely to be chosen

Variation in Data

• Data can vary due to measurement methods

Variation in Samples

• Samples can vary due to method, size

Frequency, Levels of Measurement

Levels of Measurement

• Nominal scales are qualitative, unordered: names, labels, categories, colors
• Ordinal scales are qualitative, ordered: top five ranks
• Interval scales are ordered, relatively compared, but have no absolute starting point: temperature
• Ratio scales are ordered, have absolute starting points, can be compared using ratios: sports scores

Frequency

• Frequency is the number of times a value occurs
  • Relative frequency is the ratio of the number of times a value occurs in the dataset
  • Cumulative relative frequency is the running total of relative frequencies

Experiment Design

• Explanatory variables cause change

  • Treatments are different values of the explanatory variable
  • Lurking variables affect the response variable but are not considered
  • Confounding variables are considered but cannot be distinguished from one another

• Response variables change due to explanatory variables

• Experimental units are single objects or individuals measured

• Control groups receive a placebo

  • Blinding is when the subject does not know what treatment they are receiving
  • Double blinding is when neither the experimenter nor the subject knows what treatment is being administered

• Participants should give informed consent

Sampling (Distributions) II

Continuous Random Variables

It is not possible to take the probability of any individual value within a continuous random variable’s distribution (PDF). We can only obtain the probability of a specified range within the distribution (CDF).

Example

The amount of money Josh contributes to his savings account each month is normally distributed with a mean of $55.20 and standard distribution of $8.15.

What is the probability that Josh contributes more than $60 next month?

$z$-score: $\frac{60-55.20}{8.15}=0.589$

$P(X>60)=P(z>0.589)=0.278$

NormCD(60, 100000, 8.15, 55.20)
>>> 0.2779

What amount of money would indicate the top 5% and bottom 5% of contributions?

InvNormCD(0.95, 8.15, 55.20)
>>> 68.6055
InvNormCD(0.05, 8.15, 55.20)
>>> 41.7944

Combining continuous normal distributions:

• $\mu ={\mu }_1+{\mu }_2$
• $\sigma =\sqrt{{{\sigma }_1}^2+{{\sigma }_2}^2}$

Sample Statistics

Sample statistics are point estimators of their corresponding population parameters:

• $\bar{x}\to \mu$
• $S\to \sigma$
• $\hat{p}\to p$
• $\text{sample median}\to \text{population median}$
• $\text{sample range}\to \text{population range}$

Sample statistics will never perfectly match the population. Sampling variability refers to how much estimates vary from sample to sample.

See: Chapter 7 – The Central Limit Theorem

• ${\mu }_{\hat{p}}=p$ then $\hat{p}$ is an unbiased estimator
• $n\ge 30$ samples for the CLT to apply
  • True sampling distributions account for all possible sample statistics for all samples of a given size

Sampling Distribution Model

Samples must be random (unbiased), size < 10% of population (assuming independence, have $\ge$ 10 successes/failures.

• Center: ${\mu }_{\hat{p}}=p$
• Spread: ${\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}$
• Normally distributed

Differences in Sample Proportions

• Center: ${\mu }_{\widehat{p_1}-\widehat{p_2}}=p_1-p_2$
• Spread: ${\sigma }_{\widehat{p_1}-\widehat{p_2}}=\sqrt{{{\sigma }_{p_1}}^2+{{\sigma }_{p_2}}^2}$
• Normally distributed

Example

80% of UBC students ($n=75$) and 76% of SFU students ($n=100$) passed their Calculus I finals.

Find the differences in center and spread.

${\mu }_{\widehat{p_{UBC}}-\widehat{p_{SFU}}}=0.80-0.76-0.04$

${\sigma }_{\widehat{p_{UBC}}-\widehat{p_{SFU}}}=\sqrt{\frac{0.80(0.20)}{75}+\frac{0.76(0.24)}{100}}=0.0629$

Find the probability that a sample of 100 SFU students has a higher proportion of passage than a sample of 75 students from UBC.

$\mu =\widehat{p_{UBC}}-\widehat{p_{SFU}}$

$P(\widehat{p_{SFU}}>\widehat{p_{UBC}})=P(\widehat{p_{UBC}}-\widehat{p_{SFU}}>0)$

NormCD(-100000, 0, 0.04, 0.0629)
>>> 0.262411

Differences in Sampling Distributions

• Center: ${\mu }_{{\bar{x}}_1-{\bar{x}}_2}={\mu }_1-{\mu }_2$
• Spread: ${\sigma }_{{\bar{x}}_1-{\bar{x}}_2}=\sqrt{\frac{{{\sigma }_1}^2}{n_1}+\frac{{{\sigma }_2}^2}{n_2}}$

Example

The mean weight of all male orca whales ($n=10$) follows a normal distribution with a mean of 12000 lbs and a standard deviation of 800 lbs.

The mean weight of all female orca whales ($n=15$) follows a normal distribution with a mean of 10000 lbs and a standard deviation of 900 lbs.

${\mu }_{{\bar{x}}_M-{\bar{x}}_F}=12000-10000=2000$

${\sigma }_{{\bar{x}}_M-{\bar{x}}_F}=\sqrt{\frac{{800}^2}{15}+\frac{{900}^2}{10}}=351.663$

What is the probability a sample of 15 male orcas will have a sample mean that is 3000 lbs or more than the sample mean of 10 female orcas?

$P({\bar{x}}_M+3000>{\bar{x}}_F)=P({\bar{x}}_M-{\bar{x}}_F>3000)$

NormCD(3000, 100000, 2000, 351.663)
>>> 0.002301

The occurrence of a sample statistic with a low probability causes questions about the original population parameter.