Astronomy The black hole in the M82 galaxy has a mass about 500 times the mass of the Sun. It has about the same volume as the Moon. What is the density of this black hole? mass of the Sun = 1.9891 1 0 30 kg volume of the Moon = 2.1968 10 10 k m 3

Chapter 2: Methods for Describing Sets of Data • Qualitative Data • Class frequency- the number of observations in the data set falling into a particular class/ category • Class relative frequency- the class frequency divided by the total number of observations in the data set Class relative Frequency= o Class percentage= x 100 • Visual representations o Bar graph o Pareto Diagram- bar graph in order of height o Pie Chart • Quantitative Data - Summary of Statistics: mean, standard deviation, moments, median o Visual representations § Dot Plot- horizontal axis is the scale, the dot is the data point with repeating values stacked vertically § Histogram- requires the separation into classes/ categories ▯ § Stem and Leaf- important digits are the stem, the ones are the leaf • Mean= ▯ ▯ ▯ ▯▯ (▯) ▯ ▯▯) • Median= ▯ • Skewed Data o Skewed Right- The extreme values are to the right (a tail) § Median < Mean à Mean on the right o Skewed Left- The extreme values are to the left (a tail) § Mean < Median à Mean on the left o Spread= range • Numerical Measure of Variability - Measures the central tendency in a data set o Range- o Count n X à mean X ▯ à sample variance ▯ ▯ ▯ ▯ 2 ▯▯▯▯(▯ ) ▯▯ ▯ ▯ ▯ ….▯ ▯ ▯ ▯(▯ ) o Sample Variance= S = ▯▯▯ = ▯▯▯ ▯ o Standard Deviation (s) = S ▯▯▯▯ o Z- Score/ standardized score= (gives the distance between a given measurement and the ▯ mean, expressed in standard deviations) • Chebyshev Rule- Applies to all data sets o (X- S, X + S)- tells us nothing o (X- 2S, X + 2S) – holds at least 3/4 (75% )of data o (X- 3S, X + 3S)- holds at least 8/9 (88%) of data o (X- KS, X + KS)- holds at least 1- K of data • Empirical Rule- Applies to bell- shaped data o Individually count the number within that range # of data points/ n for exact percentage o (X- S, X + S)- 68% of data o (X- 2S, X + 2S) – 95% of data o (X- 3S, X + 3S)- 99% of data • Finding Standard Deviation with only range ▯▯▯▯▯ ▯▯▯▯▯ o < S < ▯ ▯ • Percentile- the value where p% or more of your data us below and 1-p% or more of your data is above o Q L= lower quartile o Q m= median, 50th quartile o Q U= upper quartile o Inter quartile range aka IQR= Q U Q Lhis is the range of the middle 50% of your data § Definition {i} round the number inside to the nearest whole number th o If 1/2n < p< 0.5, then the pthercentile is({np}) o If 0.5

3 § Don’t use this method, use the other one above • Box Plots o o 50% of data points always lie to one side of the median Chapter 2: Probability o For A∩B= ∅ (the empty set) if P(A∪B) = P(A) + P(B) o A = all sample points not in A (A∩A DNE) à P(s)= P(A) + P(A )= 1 or P(A )= 1- P(A) o S= sample space • Counting Tricks (n choose r) Without Replacement With replacement ! Order of Pick Matters − ! Order of Pick Doesn’t Matter ! ( − + )! − ! ! − ! ! • Combinations rule ▯ ! o ( ) = ▯ ▯! ▯ ! • Additive rule of Probability • Conditional: P(A| B)=(∩) à P(b) P(A| B)= ( ∩ ) () Independence: A ∐ B if P(A| B)= P(A) or P(B| A)= P(B) o If A ∐ B then: B ∐ A and A ∐ B and C A ∐ B and A ∐ B C o If A∐B then P(A∩B)= P(A)P(B) (Vice Versa)