Summary ECON0005 Revision Grid - Statistical Methods in Economics
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Course
ECON0005 - Statistics for Economics (ECON0005)
Institution
University College London (UCL)
Book
Statistics for Business and Economics, Global Edition
Revision grid including definitions, methods and formulas covering every week's lecture notes of the ECON0005 module in an easy to navigate format for revision and exams.
Topic: Use: Formula:
The mean of a set of numbers x1,…, xk is The median of a set of numbers 𝑥1 ≤ ⋯ ≤ The mode is the most frequent outcome from a set of observations.
the sum of the set divided by its size. 𝑥𝑘 (arranged in • Typically used in the analusis of categorical data (e.g. gender,
𝑥1 + ⋯ + 𝑥𝑘 ∑𝑘𝑖=1 𝑥𝑖 ascending order) is: nationality, degree program)
= 𝑘+1
• 1 mode: unimodal
𝑘 𝑘 • If k is odd, the 𝑡ℎ observation.
2
𝑘 𝑘+2 • 2 modes: bimodal
If k is even, the average of the 2 𝑡ℎ and 2 𝑡ℎ
3+ modes: multimodal
observations.
Range The difference between the largest and the smallest observations.
• +: easy to calculate.
• -: depends only on two values in our sample. This is problematic, especially when here are extreme observations.
Population variance The average squared differences between each 𝑁
∑𝑖=1(𝑥𝑖 − 𝜇)2
2
observation and the population mean. 𝜎 =
𝑁
The sum of the squared differences between
each observation and the population mean,
divided by the population size.
Population Standard deviation The positive square root of the variance. 𝜎 = √𝜎 2
Population Coefficient of Variation Expresses dispersion as a percentage of the 𝜎
𝐶𝑉 = (𝑖𝑓 𝜇 > 0)
mean. 𝜇
𝑛
Sample variance The sum of the squared differences between 2
𝛴𝑖=1 (𝑥𝑖 − 𝑥̅ )2
each observation and the sample mean, divided 𝑠 =
𝑛−1
by the sample size minus 1.
We divide by n-1 instead of n because it
generates an unbiased estimator for the
population variance. (don’t need to understand
it now)
Frequency distributions Rule 1: Classes are non-overlapping and inclusive.
Rule 2: Define k, the number of classes.
𝑅𝑎𝑛𝑔𝑒 𝐿𝑎𝑟𝑔𝑒𝑠𝑡 𝑁𝑢𝑚𝑏𝑒𝑟−𝑆𝑚𝑎𝑙𝑙𝑒𝑠𝑡 𝑁𝑢𝑚𝑏𝑒𝑟
Rule 3: The class width is given by: =
𝑘 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑠
Histogram A histogram is a bar graph where each bar corresponds to a class and its height is proportional to the frequency in that class.
Numerical Summaries for Grouped Data: 𝑘 𝑘 1<
𝑘
∑𝑖=1 𝑓𝑖 (𝑚𝑖 − 𝜇) 2 ∑ 𝑓𝑖 𝑚𝑖 2
Population 𝑁 = ∑ 𝑓𝑖 𝑖=1 𝑓𝑖 2
𝜎2 = = − 𝜇2 = ∑ 𝑚 − 𝜇2
𝑖=1 𝑁 𝑁 𝑁 𝑖
𝑖=1
𝑘
Σ𝑖=1 𝑓𝑖 𝑚𝑖
𝜇= =
𝑁
𝑛
Sample Skewness Right skewness → Skewness > 0
Left skewness → Skewness < 0 𝑛 (𝑥 − 𝑥̅ )3
𝑆𝑘𝑒𝑤𝑛𝑒𝑠𝑠 = ∑
(𝑛 − 1)(𝑛 − 2) 𝑠3
Typically, right (left) skewness implies mean > 𝑖=1
(<) median > (<) mode.
Sample Covariance Measure of linear relationship between two ∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅ )(𝑦𝑖 − 𝑦̅)
variables. 𝑐𝑜𝑣(𝑥, 𝑦) =
𝑛−1
• 𝑐𝑜𝑣(𝑥, 𝑦) < 0 → negative linear association
• 𝑐𝑜𝑣(𝑥, 𝑦) > 0→ positive linear association
Sample Coefficient of Correlation The coefficient of correlation ranges from -1 to 𝑟𝑥𝑦 = 𝐶𝑜𝑣(𝑥, 𝑦)/𝑠𝑥 𝑠𝑦
+1 and
• rxy = 1 indicates a perfect positive linear
relationship;
• rxy = 0 indicates no linear relationship;
• rxy = -1 indicates a perfect negative
linear relationship.
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