Statistics and Data Sampling: Key Concepts for Math 1703

Population

the complete collection of "all" individuals/items to be studied.

Sample

a subcollection of members selected from the population.

Parameter

a numerical measurement describing some characteristic of a population.

Statistic

a numerical measurement describing some characteristic of a sample.

Quantitative Data

data consisting of numbers representing counts or measurements. Ex., number of pens

Discrete data

quantitative data representing a finite or countable number.

Continuous data

quantitative data representing an infinite many possible values corresponding to some continuous scale that covers a range of values without gaps, interruptions, or jumps.

Qualitative Data

data that consists of names or labels or numbers that do not represent a count or measurement.

Simple Random Sample (SRS)

a sample of n subjects selected in such a way that every possible sample of size n has the same chance of being chosen.

Systematic Sample

a sample in which the researcher selects some starting point and then selects every kth element in the population.

Stratified Sample

a sample in which the researcher subdivides the population into at least two different groups and draws a sample from each group.

Cluster Sample

a sample in which the researcher first divides the population into x number of groups and then randomly selects all members from some of those groups.

Convenience Sample

a sample in which the researcher simply uses results that are very easy to get. This is not a valid sampling method and will likely result in biased data.

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Sampling Bias

the technique used to obtain the individuals to be in the sample tends to favor one part of the population over another.

Nonresponse Bias

when individuals selected to be in the sample who do not respond to a survey have different opinions from those who do.

Response Bias

when answers on a survey do not reflect the true feelings of the respondent.

Center of Data

the measure of center is a value at the center (or middle) of a data set of numbers.

Mean

the average of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set.

Median

the middle value when a data set is ordered from least to greatest. When two values are in the middle, the median is the average of the two values.

Mode

the number that occurs most often in a data set.

Outlier

a data point that is not consistent with the bulk of the data from that group.

Percentiles

denoted by 𝑃𝑘, a value in a set of data such that 𝑘 percent of the observations are less than or equal to the value.

Quartiles

divides the data set into fourths. The 25th, 50th, and 75th percentiles denoted by 𝑄1, 𝑄2, and 𝑄3, respectively.

Bell-shaped Distribution

the highest frequency occurs in the middle and frequencies tail off to the left and the right of the middle.

Right-Skewed Distribution

the tail extends to the right of the peak longer than to the left. There are extreme values (outliers) to the right.

Left-Skewed Distribution

the tail extends to the left of the peak longer than to the right. There are extreme values (outliers) to the left.

Symmetrical Distribution

in a bell-shaped symmetrical distribution, the mean, median, and mode are approximately the same.

Right-Skewed Mean vs Median

in a right-skewed distribution the mean is larger than the median. The mean is closer to the right tail.

Left-Skewed Mean vs Median

in a left-skewed distribution the mean is smaller than the median. The mean is closer to the left tail.

Modal Class

the mode is always at the highest peak since that is where most data lie.

Resistant Measures

The median, quartiles, and IQR which can resist going toward an extreme value.

Not-Resistant Measures

The mean, range, standard deviation, and variance which are affected by extreme values.

Variation

The degree to which the data are spread out.

Range

The difference between the maximum data value and the minimum data value.

Range Formula

Range = max value - min value.

Standard Deviation

A measure of how much data values deviate from the mean.

Population Standard Deviation Formula

𝝈= √∑(𝑿𝒊−𝝁)𝟐 / 𝑵.

Sample Standard Deviation Formula

𝒔= √∑(𝒙𝒊−𝒙̅)𝟐 / (𝒏−𝟏).

Standard Deviation Properties

The value can never be negative, is zero if all values are the same, and larger values indicate greater amounts of variation.

Effect of Outliers on Standard Deviation

Outliers can drastically change the value of the standard deviation since it is a not resistant measure.

Variance

Deviation about the mean; it is the square of the standard deviation.

Population Variance Formula

𝝈𝟐= ∑(𝒙𝒊−𝝁)𝟐 / 𝑵.

Sample Variance Formula

𝒔𝟐= ∑(𝒙𝒊−𝒙̅)𝟐 / (𝒏−𝟏).

Relationship between Standard Deviation and Variance

Variance = (standard deviation)²; Standard deviation = √variance.

Z-score

The number of standard deviations that a data value 𝑥 is above or below the mean.

Z-score Characteristics

The z-score is unitless, has a mean of 0, and a standard deviation of 1.

Population Z-score Formula

𝒛= (𝒙−𝝁) / 𝝈.

General Z-score Formula

𝑧= (Variable−Mean) / Standard Deviation.

Σ (Sigma)

The sum of a set of data values.

x (Variable)

The variable usually used to represent the individual data values.

n (Sample Size)

The number of data values in a sample.

N (Population Size)

The number of data values in a population.

Sample Mean (x̅)

The mean of a set of sample values.

Population Mean (μ)

The mean of all values in a population.

If a data point is lower than the mean, it will have a (positive/negative) z-score.

negative

if a data point is higher than the mean, it will have a (positive/negative) z-score

positive

A “relatively better” z-score is the (greater/ lesser) z-score.

greater

A test was given to a statistics class. Results for 2 students are given. From the information provided, determine which student received a relatively higher grade. India scored a 79with a class average of 83 and a standard deviation of 2. Dahn scored a 75 on his exam with a class average of 80 and a standard deviation of 3

India: score = 79, class mean = 83, standard deviation = 2

Dahn: score = 75, class mean = 80, standard deviation = 3

z = (value − mean) ÷ standard deviation

then we will know that Dahn had the higher grade