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Data observation and Descriptive Statistics. Organizing Data. Frequency distribution Table that contains all the scores along with the frequency (or number of times) the score occurs. Relative frequency: proportion of the total observations included in each score. . Frequency distribution.

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organizing data
Organizing Data
  • Frequency distribution
    • Table that contains all the scores along with the frequency (or number of times) the score occurs.
    • Relative frequency: proportion of the total observations included in each score.
organizing data4
Organizing data
  • Class interval frequency distribution
    • Scores are grouped into intervals and presented along with frequency of scores in each interval.
    • Appears more organized, but does not show the exact scores within the interval.
    • To calculate the range or width of the interval:
      • (Highest score – lowest score) / # of intervals
      • Ex: 120 – 0 / 5 = 24
  • Bar graphs
    • Data that are collected on a nominal scale.
    • Qualitative variables or categorical variables.
    • Each bar represents a separate (discrete) category, and therefore, do not touch.
    • The bars on the x-axis can be placed in any order.
  • Histograms
    • To illustrate quantitative variables
      • Scores represent changes in quantity.
    • Bars touch each other and represent a variable with increasing values.
    • The values of the variable being measured have a specific order and cannot be changed.
frequency polygon
Frequency polygon
  • Line graph for quantitative variables
  • Represents continuous data: (time, age, weight)
frequency polygon11
Frequency Polygon


















descriptive statistics
Descriptive Statistics
  • Numerical measures that describe:
    • Central tendency of distribution
    • Width of distribution
    • Shape of distribution
central tendency
Central tendency
  • Describe the “middleness” of a data set
    • Mean
    • Median
    • Mode


X = ∑ X



  • Arithmetic average
  • Used for interval and ratio data
  • Formula for population mean ( µ pronounced “mu”)

µ = ∑ X



  • Formulas for sample mean
  • Not a good indicator of central tendency if distribution has extreme scores (high or low).
    • High scores pull the mean higher
    • Low scores pull the mean lower
  • Middle score of a distribution once the scores are arranged in increasing or decreasing order.
    • Used when the mean might not be a good indicator of central tendency.
    • Used with ratio, interval and ordinal data.
  • The score that occurs in the distribution with the greatest frequency.
    • Mode = 0; no mode
    • Mode = 1; unimodal
    • Mode = 2; bimodal distribution
    • Mode = 3; trimodal distribution
measures of variability
Measures of Variability
  • Range
    • From the lowest to the highest score
  • Variance
    • Average square deviation from the mean
  • Standard deviation
    • Variation from the sample mean
    • Square root of the variance
measures of variability22
Measures of Variability
  • Indicate the degree to which the scores are clustered or spread out in a distribution.
  • Ex: Two distributions of teacher to student ratio.

Which college has more variation?

  • The difference between the highest and lowest scores.
    • Provides limited information about variation.
    • Influenced by high and low scores.
    • Does not inform about variations of scores not at the extremes.
  • Examples:
    • Range = X(highest) – X (lowest)
    • College A: range = 41- 4 = 37
    • College B: range = 22-16 = 6
  • Limitations of range require a more precise way to measure variability.
  • Deviation: The degree to which the scores in a distribution vary from the mean.
  • Typical measure of variability: standard deviation (SD)
  • Variance

The first step in calculating standard deviation

  • X = Number of therapy sessions each student attended.
  • M = 4.2


Sum of deviations = 0

  • In order to eliminate negative signs, we square the deviations.
  • Sum the deviations = sum of squares or SS
  • Take the average of the SS
    • Ex: SS = 48.80
  • SD2 = Σ(X-M)2


  • That is the average of the squared deviations from the mean
  • SD2 = 9.76
standard deviation
Standard Deviation
  • Standard deviation
    • Typical amount that the scores vary or deviate from the sample mean
    • SD = Σ(X-M)2


    • That is, the square root of the variance
    • Since we take the square root, this value is now more representative of the distribution of the scores.


standard deviation29
Standard Deviation
  • X = 1, 2, 4, 4, 10
  • M = 4.2
  • SD = 3.12 (standard deviation)
  • SD2 = 9.76 (variance)
  • Always ask yourself: do these data (mean and SD) make sense based on the raw scores?
population standard deviation


σ =∑( X - µ ) ²



Population Standard Deviation
  • The average amount that the scores in a distribution vary from the mean.
  • Population standard deviation:

(σpronounced “sigma”)

sample standard deviation

σ = ∑( X - µ ) ²



Sample Standard Deviation
  • Sample is a subset of the population.
  • Use sample SD to estimate population SD.
  • Because samples are smaller than populations, there may be less variability in a sample.
  • To correct for this, we divide the sample by N – 1
    • Increases the standard deviation of the sample.
    • Provides a better estimate of population standard deviation.

s = ∑( X - X ) ²


N - 1

Unbiased Sample estimator

standard deviation

Population standard deviation

types of distributions
Types of Distributions
  • Refers to the shape of the distribution.
  • 3 types:
    • Normal distribution
    • Positively skewed distribution
    • Negatively skewed distribution
normal distribution
Normal Distribution
  • Normal distributions: Specific frequency distribution
    • Bell shaped
    • Symmetrical
    • Unimodal
  • Most distributions of variables found in nature (when samples are large) are normal distributions.
normal distribution36
Normal Distribution
  • Mean, media and mode are equal and located in the center.
skewed distributions
Skewed distributions
  • When our data are not symmetrical
    • Positively skewed distribution
    • Negatively skewed distribution

Memory hint: skew is where the tail is; also the tail looks like a skewer and it points to the skew (either positive or negative direction)

  • Kurtosis - how flat or peaked a distribution is.
  • Tall and skinny versus short and wide
    • Mesokurtic: normal
    • Leptokurtic: tall and thin
    • Platykurtic: short and fat (squatty like a platypus!)




z scores
z - Scores
  • In which country (US vs. England) is Homer Simpson considered overweight?
    • How can we make this comparison?
    • Need to convert weight in pounds and kilograms to a standardized scale.
  • Z- scores: allow for scores from different distributions to be compared under standardized conditions.
  • The need for standardization
    • Putting two different variables on the same scale
    • z-score: Transforming raw scores into standardized scores

z = (X - µ)


  • Tell us the number of standard deviations a score is from the mean.
z scores44
z- Scores
  • Class 1: M = $46.53 SD = $41.87 X = $54.76
  • Class 2: M = $53.67 SD = $18.23 X = $89.07
  • In which class did I have more money in comparison to the distribution of the other students?

Sample z-score: z = (X - M)


  • When we convert raw scores from different distributions to z-scores, these scores become part of the same z distribution and we can compare scores from different distributions.
z distribution
z Distribution
  • Characteristics: (regardless of the original distributions)
    • z score at the mean equals 0
    • Standard deviation equals 1
standard normal distribution
Standard normal distribution
  • If a z-distribution is normal, then we refer to it as a standard normal distribution.
  • Provides information about the proportion of scores that are higher or lower than any other score in the distribution.
standard normal curve table
Standard Normal Curve Table
  • Standard normal curve table (Appendix A)
  • Statisticians provided the proportion of scores that fall between any two z-scores.
  • What is the percentile rank of a z score of 1?
  • Percentile rank = proportion of scores at or below a given raw score.
    • Ex: SAT score = 1350 M = 1120 s = 340
    • 75th percentile
percentile rank
Percentile Rank

The percentage of scores that your score is higher than.

  • 89th percentile rank for height
    • You are taller than 89% of the students in the class. (you are tall!)
  • Homer Simpson: 4th percentile rank for intelligence.

he is smarter than 4% of the population (or 96% of the population is smarter than Homer).

  • GRE score: 88th percentile rank
  • Reading scores of grammar school: 18th percentile rank
  • Data organization
    • Frequency distribution, bar graph, histogram and frequency polygon.
  • Descriptive statistics
    • Central tendency = middleness of a distribution
      • Mean, median and mode
    • Measures of variation = the spread of a distribution
      • Range, standard deviation
    • Distributions can be normal or skewed (positively or negatively).
  • Z- scores
    • Method of transforming raw scores into standard scores for comparisons.
  • Normal distribution: mean z-score = 0 and standard deviation = 1
  • Normal curve table: shows the proportions of scores below the curve for a given z-score.