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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
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.