Frequency: The number of times a data point appears in a dataset.
Relative Frequency: The frequency of a particular value compared to the total number of observations, expressed as a fraction or percentage.
Class Mark: The midpoint of a class interval.
Class Boundaries: The values that define the limits of classes in grouped data.
Class Width: The difference between the upper and lower boundaries of a class.
For displaying grouped frequencies, use bar chart for clear comparisons instead of a pie chart which shows parts of a whole.
Given data: 1, 3, 5, 9, 6, 8, 3
Mean = (1 + 3 + 5 + 9 + 6 + 8 + 3)/7 = 5
Given sorted data: 1, 3, 3, 4, 5, 6, 8, 9
Median = (4 + 5) / 2 = 4.5
Given the data: 3, 4, 4, 5, 7, 8, 9, 9, 23400
Range = Maximum - Minimum = 23400 - 3 = 23397
Outlier in data affects the mean significantly more than it does the median.
Given: n = 2, S xi = 5, S xi^2 = 17
Sample Variance (s²) = (17 - 12.5) / 1 = 4.5
Sample Standard Deviation (s) = √s² = 2.121 (if calculated).
Approx. 95% of data lies within 2 standard deviations.
Approx. 99% of data lies within 3 standard deviations.
Given sorted data: 1, 3, 3, 4, 5, 6, 8, 9
Q1 (25th percentile) = 3; Q3 (75th percentile) = 7.
Extremes = Minimum and Maximum values = 1 and 9.
To draw the boxplot, plot the quartiles and extremes based on the data in question 6.
Population Mean: Average of the entire population.
Sample Mean: Average calculated from a sample of the population.
The state of birth of a group of students is (b) Qualitative data.
Given data: {10, 7, 12, 5, 7, 21, 3, 9, 9, 12, 1, 10, 13, 4}
Stem of Five:
0 | 134
0 | 57799
1 | 00223
1 | 2
Given data from Question 18: 1, 5, 9, 12, 21
Draw a boxplot using the five-number summary from question 19.
Data from question 18: Mean = 8.785714; Standard Deviation = 5.025758
In a sample of size 125, 89% of data is within 3 standard deviations, approx. 111 observations.
Possible samples (size 2) from 5 employees = 10 (5C2 = 5*4/2!)
The sample mean is not robust against outliers, as it is directly affected by their presence.
Two events A and B are independent if and only if P(A|B) = P(A).
Probability P(pass) = 1.1 is incorrect because probabilities must range from 0 to 1.
P(A or B) = P(A) + P(B) - P(A and B) = 0.5 + 0.7 - 0.3 = 0.9
A and B are not independent as P(AB) = 0.3 does not equal P(A)P(B) = 0.35.
A and B are not mutually exclusive because they can occur simultaneously.
P(Ac) = 1 - P(A) = 1 - 0.5 = 0.5
P(Ac and Bc) = 1 - P(A or B) = 0.1
Toss a coin twice: Sample Space = {H,T}
Random student selection: S = {List of students}
Random assignment of clients: Three ways as per the number of clients and salespersons.
Sum of dice to be 10, odds, etc., and probabilities when rolling two dice computed using standard methods.
For independent A and B: P(A and B) = P(A) * P(B).
Independent events cannot be mutually exclusive; they can occur together.
Probability of choosing 2 left-handed students out of 10.
5! = 120 (orderings of 5 people).
7C5 = 21 (ways to choose 5 out of 7).
Medals distribution ways: 8! / (8-3)! = 336.
Probabilities calculated for sums, evenness, etc., based on outcomes for pairs of rolled dice.
Histogram of CO2 emissions data derived from vehicle testing conducted on Sept 30, 1997.
Compare groups of vehicles using boxplots.
Plot final grades distribution in a class with Business and Engineering majors.
Events and their probabilities analyzed based on students' performance.
Stem and leaf diagrams constructed to analyze company profits, further insights provided based on gathered data.