Descriptive Stats

Descriptive Statistics: Summary Information

• Descriptive statistics provide summary information for each variable in a dataset.

• Key components include:

  • Number of cases

  • Central tendency

  • Dispersion

• Used by researchers to describe variables.

• Important caution: it generally doesn’t make sense to report the raw counts for Likert-scale responses (e.g., how many people answered 1, 2, 3, 4, or 5) in descriptive summaries; frequencies/percentages are more appropriate for categorical variables.

• Descriptive statistics lay the groundwork before conducting statistical tests that analyze differences and relationships between variables.

Data Layout and Purpose of Descriptive Statistics

• Data layout in many studies:

  • Each person’s data is on one row (case)

  • Variables are in columns

• Some variables are binary; others are interval or continuous.

• Raw data can be hard to discern patterns from; descriptive statistics help reveal patterns and summaries.

Application of Descriptive Statistics

• Reported in methods section of research reports.

• For each variable, report:

  • Mean, standard deviation (sd), range, and sample size (n or N).

  • Frequencies: how many times a particular value occurs (usually for categorical variables).

  • Percentages: describe characteristics or attributes of participants (usually for categorical variables).

Number of Cases and Sample Size

• Number of cases describes how many data points are reported.

• Denoted by n or N (depends on design).

• Typically, N is the total sample size (e.g., N = 231).

• Small n is used when there are multiple sample sizes (e.g., in experiments): n1 = 50, n2 = 60.

• Cases can be

  • People

  • Speaking turns

  • Episodes

  • Any phenomenon studied

Measures of Central Tendency

• Measures include:

  • Mean: arithmetic average; most sensitive to extreme scores.

  • Median: middle value; not sensitive to extreme scores; useful when distribution is non-normal.

  • Mode: value(s) that appear most often; useful for continuous and categorical data; some distributions have more than one mode.

• When distributions are non-normal, the median can be more informative than the mean.

• Relationship to normal curve:

  • In a perfectly normal distribution, the mean, median, and mode are equal.

  • In positively skewed distributions, mean > median.

  • In negatively skewed distributions, mean < median.

• Example connection: median home prices in Tucson might be more representative than the mean when the distribution is skewed.

Measures of Dispersion

• Describes the variability or spread of scores.

• Should be reported with the mean.

• Common dispersion measures:

  • Range: difference between highest and lowest score.

  • Standard deviation (sd): average distance of raw scores from the mean.

• Interpretations:

  • If sd = 0, all scores are the same.

  • Larger sd indicates scores differ more from the mean on average.

Crunching Numbers: Tools and Process

• Tools: calculator with square root, spreadsheet software, or statistics programs (R, SAS, SPSS).

• There are online tools offering fairly sophisticated analyses.

• Researchers must: select the appropriate descriptive statistic and the appropriate test.

• The accuracy of results depends on correct input; wrong input leads to errors.

Preliminaries: Population vs. Sample and Basic Notation

• Greek symbols typically refer to population values (e.g., µ, σ).

• Sample estimates (often denoted by non-Greek or accented symbols; see below) are used to estimate population values.

• In most studies, we cannot measure everyone in the population; we estimate population values from a sample.

• The symbol Σ (capital sigma) means “sum everything that follows.”

• Normal mathematics (PEMDAS) applies to statistical formulas:

  • Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

Population vs. Sample Symbols

• Common symbolic mapping (as presented):

  • Population: mean = µ, variance = σ^2, standard deviation = σ

  • Sample: mean = ar{x}, variance = s^2, standard deviation = s

• (The slide row lists a term-by-symbol mapping; focus on the standard convention above.)

General Variance Formula

• Variance measures the average squared deviation from the mean.

• Two main formulas depending on whether you describe the population or estimate it from a sample:

  • Population variance: $ext{Var} = rac{1}{N} \, ext{or} \, rac{1}{N} \, ext{sum}<em>{i=1}^N (x</em>i - \mu)^2$

  • Sample variance: $s^2 = rac{1}{n-1} \, ext{sum}<em>{i=1}^n (x</em>i - \bar{x})^2$

• Notation and order of operations matter; (capital sigma) sums deviations, then divide by the appropriate denominator (N for population, N-1 for sample in most cases).

Computing Variance: An Example

• The dataset (livesup) provides

  • sum of squared deviations = 11.6

  • sum of deviations from mean before squaring = not shown, but implied by the data

  • mean = 3.6 (from the slide): the average score.

• Denominators under discussion: 14 or 15, depending on whether you use the sample or population formula.

• Possible variance values:

  • 11.6 / 14 = 0.83

  • 11.6 / 15 = 0.77

• The slide notes that 0.77 appears as the average of the squared deviations (dividing by N) and is the population variance formula.

• The slide also notes the difference between the two formulas is small for large samples.

Standard Deviation and Its Interpretation in the Example

• Standard deviation is the square root of the variance:

  • If variance = 0.83 (sample formula): $s = \sqrt{0.83} \approx 0.91$

  • If variance = 0.77 (alternative, population-based denominator): $s = \sqrt{0.77} \approx 0.88$

• The slide suggests reporting the standard deviation with the chosen variance denominator and notes that differences are negligible for large samples.

Implications of Variance and Standard Deviation

• Variance tells us about the relationship of a score to the rest of the scores.

• Standard deviation ranges from zero to the maximum variability possible for a given scale.

• A score’s percentile relative to the mean can be inferred from the distribution (e.g., SAT percentile logic).

• Knowing variance helps predict where most scores lie and aids in hypothesis testing by generalizing from sample to population (lead-in to next lecture).

The Normal Curve: Theoretical vs Empirical

• Theoretical Normal Distribution (bell curve):

  • Y-axis: Relative Frequency

  • Area under the curve equals 1 (or 100%)

  • Most data cluster around the center; symmetrical; tails extend to infinity in both directions

  • Mean = Median = Mode

• Empirical Normal Distribution: an approximation to the theoretical curve; becomes closer to normal with larger samples; based on sampling simulations.

The Normal Curve: Percentages Within Standard Deviation

• Known percentages around the mean on a normal curve:

  • 34% of scores lie between the mean and +1 SD (or between mean and -1 SD).

  • An additional 14% lie between +1 SD and +2 SD (or between -1 SD and -2 SD).

  • Therefore, about 96% of all scores are within ±2 SDs of the mean (34% + 34% + 14% + 14%).

• These 34% and 14% values are often used for quick mental checks (e.g., Mean Height = 65, SD = 4 as a mnemonic example).

The Standard Normal Curve and Z-Scores

• A standard score expresses a score’s position relative to the mean using the standard deviation as the unit; it standardizes different scales.

• A z-score represents the number of standard deviations a value is above or below the mean:

  • Population-based standardization: $z = rac{x - \mu}{\sigma}$

  • Standard normal distribution has mean 0 and standard deviation 1: $\mu<em>Z = 0, \\sigma</em>Z = 1$

• Converting to z-scores allows comparison across differently scaled variables.

• Note: The z-table is applicable only if the data are (approximately) normally distributed.

The Normal Curve Table and Percentiles

• The normal curve table provides the precise percentage of scores between the mean (z = 0) and any z-score.

• Uses:

  • Proportion above or below a given z-score

  • Proportion between the mean and a given z-score

  • Proportion between two z-scores

• When using z-tables, remember they assume a normal distribution; non-normal distributions (rectangular, skewed, leptokurtic, bimodal) do not align with the standard normal table.

Quick Visuals and External Tools

• A common demonstration site shows areas under the standard normal curve (e.g., areas between 0 and 1, or 0 and -1) and the corresponding decimal areas.

• The Z-table can also be used to determine z-scores for a given proportion of scores and vice versa.

Finding Area When the Score is Known

• Step-by-step approach:
  1) Convert the raw score to a z-score.
  2) Draw the normal curve and locate the z-score.
  3) Shade the area corresponding to the desired proportion.
  4) Make a rough estimate of the shaded area’s percentage.
  5) Use the normal curve table to find the exact percentage.
  6) Check that the exact percentage is close to the estimate.

• Example (mean = 10, sd = 2):

  • Find the percentage above 12: 16%

  • Find the percentage below 12: 84%

  • Find the percentage above 8: 84%

  • Find the percentage below 8: 16%

  • Find the percentage above 9: 69%

  • Find the percentage below 7: 7%

Finding Area Between Two Scores

• Steps identical to the above, but shade the area between two z-scores.

• Example (mean = 10, sd = 2):

  • Between 10 and 12: 34%

  • Between 8 and 10: 34%

  • Between 6 and 10: 48%

  • Between 6 and 8: 14%

  • Between 10.5 and 11: 10%

  • Between 8.5 and 11: 46%

Finding Scores When the Area (z-score) is Known

• Steps:
  1) Draw the normal curve; shade the approximate area corresponding to the desired percentage.
  2) Make a rough estimate of the starting z-score where the shaded area begins.
  3) Use the normal curve table to find the exact z-score.
  4) Convert the z-score to a raw score if desired: $x = z \, \cdot \, s + \bar{x}$ or more generally: $x = z\cdot s + \bar{x}$

• Example scaffolding (mean = 10, sd = 2):

  • What raw score corresponds to certain percentile positions (e.g., 50% above, 84% below, 98% above, 62% below, 30% above).

• Practical tip: for the last two examples, solving for z may require working backward; online z-tables can be helpful.

Remember: Normality and the Z-Table

• Key reminder: The standard normal distribution table (z-table) should be used only when the score distribution is normal.

• If the distribution differs markedly from normality (non-normal), transforming to z-scores and using the z-table is inappropriate for exact probabilities.

• Non-normal distributions include rectangular, skewed, leptokurtic, and bimodal shapes.

Standard Error Revisited and Confidence Intervals for the Mean

• The standard error estimates how similar a sample mean is to the population mean.

• Confidence interval for the mean (commonly 95%):

  • Example: sample of 100 students, mean attitude score toward tuition increases = 4 (on a 1–10 scale), sample sd = 1.

  • 95% CI for the population mean: $\bar{x} \pm z_{\alpha/2} \cdot \frac{s}{\sqrt{n}} = 4 \pm 1.96 \cdot \frac{1}{\sqrt{100}} = 4 \pm 0.196$

  • Resulting interval: $[3.804, 4.196]$

  • Interpretation: 5% of the time the population mean would fall outside this interval.

Our Friend, 1.96: The 95% Confidence Multiplier

• On a standard normal distribution, approximately 5% of the curve lies outside ±1.96 standard errors from the mean.

• Therefore, ±1.96 captures the central 95% of the distribution around the mean in confident interval estimation.

Degrees of Freedom (DF)

• Intuition: Degrees of freedom are the number of independent pieces of information available to estimate a parameter.

• Concrete example: with two observations, you have two independent observations for the mean; however, when calculating the variance, only one independent piece of information remains because the two observations are constrained by their distance from the mean.

• Conceptual view: DF reflect how many values are free to vary when estimating a parameter given fixed other parameters.

• Practical note: In many cases, using DF properly prevents underestimating the population parameter when using sample estimates.

• Summary: A dataset’s sample is an estimate of the population; neglecting degrees of freedom can bias parameter estimates.