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Fundamentals of Statistics and Data Analysis
Jul 31, 2024
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Statistics Lecture Notes
Key Concepts
Mean
Median
Mode
Range
Quartiles and Interquartile Range (IQR)
Outliers
Box and Whisker Plot
Skewness
Dot Plot
Stem-and-Leaf Plot
Frequency Table
Histogram
Percentiles
Data Set Analysis
Example Data Set 1: {7, 7, 10, 14, 15, 23, 32}
Mean
:
Sum: 108
Mean = 108 / 7 = 15.43
Median
:
Middle number: 14
Mode
:
Most frequent number: 7
Range
:
Difference between highest and lowest number: 32 - 7 = 25
Example Data Set 2: {11, 15, 15, 21, 37, 41, 59, 59}
Mean
:
Sum: 258
Mean = 258 / 8 = 32.25
Median
:
Middle numbers: 21 and 37
Median = (21 + 37) / 2 = 29
Mode
:
Numbers that appear most frequently: 15 and 59 (Bimodal)
Range
:
Difference between highest and lowest number: 59 - 11 = 48
Quartiles and Interquartile Range (IQR)
Quartiles
: Dividing a data set into four equal parts
Q1
: Median of the lower half
Q2
: Median of the entire data set
Q3
: Median of the upper half
IQR
: Difference between Q3 and Q1 (IQR = Q3 - Q1)
Determining Outliers
:
Not an outlier if within range:
Q1 - 1.5 * IQR to Q3 + 1.5 * IQR
Example Data Set: {7, 11, 14, 5, 8, 27, 16, 10, 13, 17, 16}
Q1
: 8
Q2
: 13
Q3
: 16
IQR
: Q3 - Q1 = 16 - 8 = 8
Outliers
:
Range: -4 to 28
27 is not an outlier
Box and Whisker Plot
Components
:
Minimum
Q1
Median (Q2)
Q3
Maximum
Outliers
: Represented by points outside the main box
Skewness
Symmetric Distribution
: Mean = Median
Skewed Right (Positive Skew)
:
Mean > Median
Box and whisker plot: Right side longer
Skewed Left (Negative Skew)
:
Mean < Median
Box and whisker plot: Left side longer
Dot Plot
Example: Data set {5, 8, 3, 7, 1, 5, 3, 2, 3, 3, 8, 5}
Dot plot shows frequency of each number
Mode is the number with the most dots
Stem-and-Leaf Plot
Construction
:
Separate each number into a stem and leaf
Example: 78 -> Stem: 7, Leaf: 8
Key
: Provides explanation of stem-leaf representation
Decimal Values Representation
: Similar method, but stems can represent integer part and leaves decimal part
Frequency Table
Components
:
Value
Frequency
Relative Frequency
Cumulative Relative Frequency
Example Calculation
:
Data set: {5, 9, 8, 7, 8, 12, 9, 8, 10, 8, 9, 7}
Frequency of each value found
Relative Frequency: Frequency / Total number of data points
Cumulative Relative Frequency: Running total of the relative frequencies
Mean Calculation from Frequency Table
: Sum of (value * frequency) / Total frequency
Histogram
Construction
:
Bar graph representation of data
Bars are connected
Example: Test scores categorized by grade intervals
Percentiles
Calculation
: Using cumulative relative frequency to determine percentile rank
Example
:
Data set with frequency and cumulative relative frequency columns
60th percentile: Average of values around 0.60 cumulative relative frequency
80th percentile: Average of values around 0.80 cumulative relative frequency
Summary
Statistics Basics
: Mean, median, mode, range, quartiles, IQR, outliers, skewness, dot plots, stem-and-leaf plots, frequency tables, histograms, percentiles
Importance
: Understanding these concepts is crucial for analyzing and interpreting data effectively
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