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Introductory Statistics Key Concepts

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  • What is the formula for the population mean?

    The population mean \(\mu\) is calculated as the sum of all values divided by the population size: \(\mu = \frac{\sum_{i=1}^N x_i}{N}\).

  • How is the sample mean calculated?

    The sample mean \(\bar{x}\) is the sum of all sample values divided by the sample size: \(\bar{x} = \frac{\sum_{i=1}^n x_i}{n}\).

  • What is the formula for the population variance?

    Population variance \(\sigma^2\) is the average of squared differences from the mean: \(\sigma^2 = \frac{\sum_{i=1}^N (x_i - \mu)^2}{N}\).

  • How do you calculate the sample variance?

    Sample variance \(s^2\) is the adjusted average of squared differences from the sample mean: \(s^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n-1}\).

  • What is the standard deviation and how is it related to variance?

    Standard deviation is the square root of the variance, providing a measure of spread in the same units as the data.

  • What does skewness measure in a distribution?

    Skewness measures the asymmetry of a distribution around the mean: zero skewness is symmetric, positive skewness indicates a right tail, and negative skewness indicates a left tail.

  • Describe a symmetric (not skewed) distribution.

    A symmetric distribution has skewness = 0, and the mean equals the median.

  • What characterizes a moderately right skewed distribution?

    Moderately right skewed distributions have skewness > 0, with the mean usually greater than the median.

  • What characterizes a moderately left skewed distribution?

    Moderately left skewed distributions have skewness < 0, with the mean usually less than the median.

  • What is kurtosis and what does it indicate?

    Kurtosis measures the heaviness of a distribution's tails relative to its center, indicating the likelihood of extreme values.

  • Define leptokurtic, mesokurtic, and platykurtic distributions.

    Leptokurtic: kurtosis > 3 (heavy tails), Mesokurtic: kurtosis = 3 (normal), Platykurtic: kurtosis < 3 (light tails).

  • What is a box plot and what does it show?

    A box plot graphically displays the five-number summary: minimum, Q1, median, Q3, and maximum, highlighting data spread and outliers.

  • What is the empirical rule for normal distributions?

    Approximately 68% of data lie within 1 standard deviation, 95% within 2, and 99.7% within 3 standard deviations of the mean.

  • What is the standard normal distribution?

    A normal distribution with mean 0 and standard deviation 1, used to calculate probabilities for standardized values (z-scores).

  • How is a z-score interpreted?

    A z-score indicates how many standard deviations a value is from the mean: z=0 at mean, z>0 above mean, z<0 below mean.

  • What is the formula for the z-score of a sample mean?

    \(z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}\), where \(\bar{x}\) is sample mean, \(\mu\) population mean, \(\sigma\) population standard deviation, and \(n\) sample size.

  • What is the central limit theorem?

    The sampling distribution of the sample mean is approximately normal if the population is normal or the sample size is large (usually n>30).

  • What is the formula for the standard error of the mean?

    Standard error = \(\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is population standard deviation and \(n\) is sample size.

  • What are the properties of a binomial distribution?

    Fixed number of independent trials, two outcomes per trial (success/failure), constant probability of success \(\pi\), and the random variable counts number of successes.

  • When can a binomial distribution be approximated by a normal distribution?

    When both \(n\pi \geq 10\) and \(n(1-\pi) \geq 10\) are satisfied, the binomial distribution is approximately normal.