Lesson 2.2: Numerically Summarizing Data

Question 1

population mean

Answer 1

• denoted by 𝜇
• population size = 𝑁

Question 2

sample mean

Answer 2

• denoted by 𝑥 with line over it
• sample size = 𝑛
• R: mean(DataTable$Column)

Question 3

median

Answer 3

• middle number of data in ascending order
• if odd number of observaions: middle number
• if even number of observations: mean of two middle numbers
• not greatly affected by outliers
• R: median(DataTable$Column)

Question 4

mode

Answer 4

• most frequent observation
• R: Mode <- function(x) {
  ux unique(x)
  ux [which.max (tabulate(match(x, ux)))]
  }
• histogram: hist (data,breaks=seq (low, high, interval))

Question 5

Range
(R)

Answer 5

• difference between largest and smallest value
  -** R: range(DataTable$Column)** shows lowest and highest values

Question 6

population variance

Answer 6

• denoted by σ²
• sum of the squared deviations about
  the population mean (x-μ)²
• Divided by number of observations in population (N)

> (n=length(data$Score))
(x =data$Score)
(v.population = sum((x-mean(x))^2) / n)

Question 7

sample variance

Answer 7

• denoted by s²
• sum of the squared deviations about
  the sample mean (x-sample mean)²
• Divided by number of observations in sample minus 1 (n-1)

R: var(DataTable$Column)
R var() only gives sample variance

Question 8

standard deviation

population and sample

Answer 8

square root of the variance
- denoted by σ (population) or s (sample)

Question 9

Coefficient of Variation
(CV)

Answer 9

• measures the scatter in the data relative to mean
• CV = standard deviation (s) / sample mean (x)
• always expressed as percentage

Question 10

Computing standard deviation

Excel and R

Answer 10

• Excel: = STDEV(data range)
• R: sd(vector)

Question 11

Empirical Rule

Answer 11

• Approximately 68% data between 𝜇−1𝜎 𝑎𝑛𝑑 𝜇+1𝜎
• Approximately 95% data between 𝜇−2𝜎 𝑎𝑛𝑑 𝜇+2𝜎
• Approximately 99.7% data between 𝜇−3𝜎 𝑎𝑛𝑑 𝜇+3𝜎

left to right percentages under curve:
0.15 + (2.35 + (13.5 + (34 + 34) + 13.5) + 2.35) + 0.15

Question 12

Standardized values

z value

Answer 12

• Compute Probabilities
• Compare two different distributions
• We compute standardized values
• z = 2 : Data value is 2 standard deviations above the mean
• z = -1.6 : Data value is 1.6 standard deviations below the mean

z = (Data value (y) - mean(μ)) / Standard deviation (σ)

Question 13

p-value

Answer 13

• Represents the area under the normal distribution curve towards left side
• 1.0 z-value = 0.15 + 2.35 + 13.50 + 34.00 + 34.00 = 84% p-value (0.8413)
• area under normal distribution curve = 1

Question 14

p-value calculation for specific range

example

Answer 14

probability of student scoring b/n 450 and 600 on SAT
mean = 500, sd = 100
z = (600-500)/100 = 1.0 = p-value 0.8413
z = (450-500)/100 = -0.50 = p-value 0.3085
0.8413 - 0.3085 = 0.5328 or 53.28%

Question 15

Standard normal curve
(𝜇=0, 𝜎=1)

Answer 15

• Symmetric about its mean 𝜇=0,𝜎=1
• Mean = Median = Mode
• Single peak at z=0
• Inflection point at −1𝑎𝑛𝑑+1
• Area under the curve = 1
• Area of left ( 𝑚𝑒𝑎𝑛 𝜇=0) = Area of right = ½
• Follows the Empirical Rule

Question 16

Normalizing Data

Computing z-values

Excel and R

Answer 16

• Excel: =STANDARDIZE(data value, mean, sd)
• R: scale(data, mean, sd)

Question 17

Computing p-values
𝑃(𝑧)𝑥 < 𝑧

Excel and R

Answer 17

‘Left’ Area (Probability) under Standard Normal Curve
- Excel: =NORMSDIST(z-value) Normal standard dist
- R: pnorm(z-value)

‘Right’ Area (Probability) under Standard Normal Curve
- Excel: =1-NORMSDIST(z-value)
- R: 1 - pnorm(z-value)

‘In Between’ Area (Probability) under Standard Normal
Curve
- Excel: =NORMSDIST(high z) - NORMSDIST(low z)
- R: pnorm(high z) - pnorm(low z)

Question 18

Converting p-value (‘left’ area) to z-value

Excel and R

Answer 18

• Excel: =NORMSINV(p-value)
• R: qnorm(p-value)

Question 19

Skewness

types (3)

Answer 19

• Mean < Median < Mode: Negative / left skewed distribution
• Mean = Median = Mode: Symmetrical distribution with zero skewness
• Mean > Median > Mode: Positive / right skewed distribution

Question 20

Percentile

Answer 20

• The k^th percentile of a set of data is a value such that k percent of the observations are less than or equal to the value
• eg. P2 = 2% of observations are <= value

Question 21

Quartile

Answer 21

The quartiles divide the data into 4 equal parts

*First quartile: Q1 *
- Bottom 25% (25 percentile)

Second quartile: Q2
-Bottom 50% = 50 percentile (median)

Third quartile: Q3
- Bottom 75% = 75 percentile

Question 22

Boxplot

features and R command

Answer 22

• Line = median
• box = Q1 and Q3
• viscus (brackets) = min and max
• dot = outlier
• R: boxplot(data vector)

Question 23

Skewness and Boxplots

Answer 23

Normal Distribution
- (Q3-Q2) = (Q2-Q1)

Positive Skew
- (Q3-Q2) > (Q2-Q1)

Negative Skew
- (Q3-Q2) < (Q2-Q1)

Question 24

Data Standardization and Scaling

Answer 24

Standardization Data Variation (z-value)
- Range: -3 to +3

**Scaling Data Variation **
- (value - min value) / (max value - min value)
- Range: 0 to 1
- not effective with outliers bc will suppress scaling values of other data elements