Exam Notes on Standard Deviation, Distributions, and Z-Scores

General Properties of Standard Deviation

• Measures spread (variability) by looking at how far the observations are from their mean.
• $S = 0$ indicates no spread; all scores are the same.
• $S$ increases as scores become more spread out (further from the mean) indicating more variation.
• $S$, like the mean, is influenced by extreme values (non-resistant).
• Context varies typically by standard deviation (SD) from the mean.
  • Example: The height of power forwards in the NBA typically varies by 1.52 inches from the mean of 80.1 inches.

Distributions and Skewness

• Symmetric Distribution:
  • Mean and median are approximately equal.
• Skewed Right Distribution:
  • Pulled towards extreme values.
  • Mean > Median.
  • Center: Median.
  • Spread: Range or IQR.
• Skewed Left Distribution:
  • Mean < Median.
  • Center: Median.
  • Spread: Range or IQR.
  • The median is a better representation of center in skewed distributions.

Data Types and Visualizations

• Categorical Data:
  • Pie charts, bar graphs.
• Quantitative Data:
  • Dot plots, stem plots, histograms.

Frequency Tables

• Counts, relative frequency, marginal frequency, joint frequency, and conditional frequencies.

Measures of Center and Spread

• Mean: Symmetrical data.
• Standard Deviation: Spread, typical distance from the mean.

Calculations and Interpretations

• Five-number summary.
• Determining outliers.
• Describing and comparing distributions.

Median and Outliers

• Median is the 50th percentile.
• When outliers are removed, the mean decreases if the outlier was larger than the mean and increases if the outlier was smaller than the mean.
• If a value the same as the mean is added, the SD does not change.
• Adding a number higher but close to the mean, the spread decreases, and the mean increases because the new value is higher than the current mean.
• By adding an outlier far away from the mean, the mean increases because it is pulled towards the data, and the SD would increase as well due to the non-resistance.

Interquartile Range (IQR)

• Identifies the middle 50% of the data.
• $IQR = Q3 - Q1$
• Outlier Identification:
  • Greater than $Q3 + 1.5 \times IQR$
  • Less than $Q1 - 1.5 \times IQR$

Example

• Data set: $(1, 10, 20, 30, 40, 50, 120)$
• Median (Q2) = 30
• Q1 = 10
• Q3 = 50
• $IQR = 50 - 10 = 40$
• Lower outlier boundary: $10 - 1.5(40) = -50$
• Higher outlier boundary: $50 + 1.5(40) = 110$
• 120 is an outlier.

Five-Number Summary

• Minimum, Q1, Median (Q2), Q3, Maximum.
• Displayed on boxplots.
• Include outliers.
• Needs title and label.

Statistical Calculations

• Using stats plot in calculator to generate the five-number summary.
• Accessing mean ($\bar{x}$) using calculator functions.
• $n-1$ describes characteristics of samples.

Parameter vs. Statistic

• Parameter: Describes some characteristic of a population.
• Statistic: Describes some characteristic of a sample.

Variance

• Average squared deviation from the mean.
• $s^2 = \frac{\sum(x - \bar{x})^2}{n-1}$
  • Where:
    • $x$ is each value in the data set.
    • $\bar{x}$ is the mean.
    • $n$ is the number of values in the data set.

Standard Deviation

• Square root of the average squared deviation from the mean (square root of variance).
• If not squared, data can cancel to zero, showing no variance.
• Expressed in the same units as the original data.

Standard Deviation Types

• $s<em>x$ and $\sigma</em>x$ are standard deviations for samples and populations, respectively.

Percentiles and Relative Location

• The $p^{th}$ percentile of a distribution is the value with $p$ percent of the observations less than or equal to it.
  • Example: If Jenny is at the 88th percentile in a class's test score, it means that 88% of scores are below or equal to Jenny's score.

Cumulative Relative Frequency Graph

• Displays the cumulative relative frequency of each class of a frequency distribution.

Z-Scores

• How many standard deviations from the mean an observation falls and in what direction.
• $z = \frac{x - \mu}{\sigma}$
  • $x$ is the observed value.
  • $\mu$ is the mean.
  • $\sigma$ is the standard deviation.
• Units of measure for a z-score are standard deviations (SD above or below the mean).

Transformations

• Adding or subtracting a constant:
  • Affects measures of center and location (mean, median, quartiles, percentiles).
  • Does not change shape and measures of spread (range, IQR, SD).
• Multiplying or dividing each observation by the same number:
  • Affects measures of center and location.
  • Affects measures of spread.
  • Does not change the shape of the distribution.

Percentiles

• Measure of relative location, described as "at" a certain percentile, not "in".

Standardized Score

• Converting to a comparable number.
• Number of standard deviations above/below the mean.

Standard Normal Distribution

• Mean: 0
• SD: 1

Transforming Data

• Adding/subtracting a constant $a$:
  • Shape: Unchanged
  • Center: $+ a$
  • Spread: Unchanged
• Multiplying/dividing by a constant $b$:
  • Shape: Unchanged
  • Center: $\times / \div b$
  • Spread: $\times / \div b$

Density Curves

• Always on or above the horizontal axis.
• Has area exactly 1 underneath it regardless of mean and SD.
• Describes the overall pattern of a distribution.
• The area under the curve and above any interval of values on the x-axis is the proportion of all observations that fall in the interval.
• Can have different SD and mean from real distribution but it is closed (an idealized description).

Normal Distribution

• Always symmetric, single-peaked, and bell-shaped.
• Any specific normal curve is completely described by giving its mean $\mu$ and standard deviation $\sigma$: $N(\mu, \sigma)$

The Empirical Rule (68-95-99.7 Rule)

• 68% of data falls within 1 standard deviation of the mean.
• 95% of data falls within 2 standard deviations of the mean.
• 99.7% of data falls within 3 standard deviations of the mean.

Standard Normal Distribution

• $N(0, 1)$

Calculations with Normal Distributions

• To find percentile for a score:
  • Calculate the z-score.
  • Find the area to the left of the z-score on the standard normal table.

Assessing Normality

• If a plot of scores against expected normal scores is a straight line, the data can be considered approximately normal.

Calculations and Z-Scores

• Always draw a normal curve.
• Make calculations and use z-scores to find the area out of the table.