Flashcards · 10 cards

Statistics & correlation

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Rule
The mean and standard deviation using an assumed mean $A$ ?
Answer
With $d = x - A$ : $\bar x = A + \dfrac{\sum fd}{\sum f}$ and $\sigma = \sqrt{\dfrac{\sum fd^2}{\sum f} - \left(\dfrac{\sum fd}{\sum f}\right)^2}$ .
Deviations from an assumed meanMean = A + Σd ÷ n
From the lesson: Mean and standard deviation by the assumed mean
Know it
Why subtract $\left(\dfrac{\sum fd}{\sum f}\right)^2$ in the standard deviation?
Answer
The deviations are measured from $A$ , not from the mean. The term is zero only when $A$ happens to be the mean.
From the lesson: Mean and standard deviation by the assumed mean
Rule
Bar heights in a histogram with unequal classes?
Answer
Frequency density = frequency ÷ class width, so each bar's area is its frequency.
Frequency densityHeight = frequency ÷ class width, so area = frequency
From the lesson: Histograms, the mode and ogives
Rule
How do you read the mode from a histogram?
Answer
From the top corners of the tallest bar, draw two lines crossing to the tops of its neighbours. The mode is directly below where they cross.
Reading the modeThe crossed lines lean towards the taller neighbour
From the lesson: Histograms, the mode and ogives
Rule
Plotting and reading a cumulative frequency curve?
Answer
Plot each running total at the upper class boundary. Read across from $\frac14 N$ , $\frac12 N$ and $\frac34 N$ , then down.
Reading the curveAcross from ¼N, ½N and ¾N, then down
From the lesson: Histograms, the mode and ogives
Rule
Spearman's rank correlation coefficient?
Answer
$\rho = 1 - \dfrac{6\sum d^2}{n(n^2 - 1)}$ , where $d$ is the difference between the two ranks.
Spearman's coefficientρ = 1 − 6Σd² ÷ n(n² − 1)
From the lesson: Correlation and lines of best fit
Know it
What does the value of $\rho$ tell you?
Answer
It lies between $-1$ and $1$ . Close to 1: the rankings agree. Close to $-1$ : they are reversed. Close to 0: little relationship.
From the lesson: Correlation and lines of best fit
Know it
The table gives marks. What do you put into Spearman's formula?
Answer
Ranks, not marks: rank each set first (1 for the highest). Subtract the ranks, then square.
From the lesson: Correlation and lines of best fit
Rule
Which point must a line of best fit pass through?
Answer
The mean point $(\bar x, \bar y)$ .
Line of best fitThrough the mean point (x̄, ȳ)
From the lesson: Correlation and lines of best fit
Which method?
WAEC 2013 · Paper 2 · Q6

Age (years) 12–14 15–17 18–20 21–23 24–26

Frequency 6 10 3 2 1

The table shows the distribution of ages of 22 students in a school. Using an assumed mean of 19, calculate, correct to three significant figures, the:
mean age;
standard deviation of the distribution.
What are the class marks and the values of $d$ ?
Answer
Class marks 13, 16, 19, 22, 25. With $A = 19$ , $d = -6, -3, 0, 3, 6$ . Then find $\sum fd$ and $\sum fd^2$ .
From the lesson: Mean and standard deviation by the assumed mean Try the question