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

Amount f(frequency) rf(relative frequency)

$0.00 2 0.125

$0.13 1 0.0625

$0.93 1 0.0625

$1.00 1 0.0625

$10.00 1 0.0625

$32.00 1 0.0625

$45.53 1 0.0625

$56.00 1 0.0625

$60.00 1 0.0625

$63.25 1 0.0625

$74.93 1 0.0625

$80.00 1 0.0625

$85.28 1 0.0625

$115.35 1 0.0625

$120.00 1 0.0625

n=16 1.00

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

Class interval frequency distribution

Class interval f (frequency) rf ( relative frequency)

$0-$24 6 .375

$25-$48 2 .125

$49-$73 3 .1875

$74-$98 3 .1875

$99-$124 2 .125

n = 16 1.00

Graphs

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.

Bar Graph

Graphs

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.

Histogram

Frequency polygon

Line graph for quantitative variables Represents continuous data: (time, age, weight)

Frequency Polygon

AGE 22.06 24.0525.04 25.04 25.07 25.07 26.03 26.11 27.03 27.11 29.03 29.05 29.05 34 37.1 53

Descriptive Statistics

Numerical measures that describe: Central tendency of distribution Width of distribution Shape of distribution

Central tendency

Describe the “middleness” of a data set Mean Median Mode

Mean Arithmetic average Used for interval and ratio data

Formula for population mean ( µ pronounced “mu”)

µ = ∑ X _____ N

Formulas for sample mean

_ X = ∑ X _____ n

Mean

Amount f(frequency) rf(relative frequency)$0.00 2 0.125$0.13 1 0.0625$0.93 1 0.0625$1.00 1 0.0625

$10.00 1 0.0625$32.00 1 0.0625$45.53 1 0.0625$56.00 1 0.0625$60.00 1 0.0625$63.25 1 0.0625$74.93 1 0.0625$80.00 1 0.0625$85.28 1 0.0625$115.35 1 0.0625$120.00 1 0.0625

$46.53 n=16 1

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

Median

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.

Median

$0.00$0.00$0.13$0.93$1.00

$10.00$32.00$45.53$56.00$60.00$63.25$74.93$80.00$85.28

$115.35$120.00

Mode

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

Mode

Amount f(frequency)rf(relative

frequency)$0.00 2 0.125$0.13 1 0.0625$0.93 1 0.0625$1.00 1 0.0625

$10.00 1 0.0625$32.00 1 0.0625$45.53 1 0.0625$56.00 1 0.0625$60.00 1 0.0625$63.25 1 0.0625$74.93 1 0.0625$80.00 1 0.0625$85.28 1 0.0625

$115.35 1 0.0625$120.00 1 0.0625

$46.53 n=16 1

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 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?

College A College B

4 16

12 19

41 22

Sum = 57 Sum = 57

Mean = 19

Mean = 19

Range

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

Variance

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)

VarianceThe first step in calculating standard deviation

Variance

X = Number of therapy sessions each student attended.

M = 4.2 “Deviation”

Sum of deviations = 0

Variance

In order to eliminate negative signs, we square the deviations.

Sum the deviations = sum of squares or SS

Variance

Take the average of the SS Ex: SS = 48.80

SD2 = Σ(X-M)2

N That is the average of the squared deviations from the

mean

SD2 = 9.76

Standard Deviation

Standard deviation Typical amount that the scores vary or deviate

from the sample mean

SD = Σ(X-M)2

N

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 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

The average amount that the scores in a distribution vary from the mean.

Population standard deviation: (σ pronounced “sigma”) √

____ σ = ∑( X - µ ) ² _________ N

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.

σ = ∑( X - µ ) ² _________ N

Unbiased Sample estimator standard deviation

Population standard deviation

√ s = ∑( X - X ) ² _________ N - 1

√

Sample Standard Deviation

X X - mean X - mean squared$0.00 -$46.53 $2,165.04$0.00 -$46.53 $2,165.04$0.13 -$46.40 $2,152.96$0.93 -$45.60 $2,079.36$1.00 -$45.53 $2,072.98$10.00 -$36.53 $1,334.44$32.00 -$14.53 $211.12$45.53 -$1.00 $1.00$56.00 $9.47 $89.68$60.00 $13.47 $181.44$63.25 $16.72 $279.56$74.93 $28.40 $806.56$80.00 $33.47 $1,120.24$85.28 $38.75 $1,501.56

$115.35 $68.82 $4,736.19$120.00 $73.47 $5,397.84

$46.53 N = 16 SS = $26,295.02

Types of Distributions

Refers to the shape of the distribution. 3 types:

Normal distribution Positively skewed distribution Negatively skewed 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 Distribution

Mean, media and mode are equal and located in the center.

Normal Distribution

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)

Skewed Distributions

Kurtosis

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!)

Kurtosis

leptokurtic

mesokurtic

platykurtic

Skewness, Number of Modes, and Kurtosis in Distribution of Housing Prices

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) σ

Tell us the number of standard deviations a score is from the mean.

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) s

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

Characteristics: (regardless of the original distributions) z score at the mean equals 0 Standard deviation equals 1

z distribution of exam scores

M = 70s = 10

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 (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

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

Review 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.