11.1 Descriptive statistics for survey data

Descriptive statistics are essential tools for understanding survey data. They help summarize and interpret large datasets, revealing key patterns and trends. From measures of central tendency to data visualization techniques, these methods provide valuable insights into the characteristics of survey responses.

Exploring central tendency, dispersion, and distribution shapes allows researchers to paint a comprehensive picture of their data. By using various statistical measures and graphical representations, analysts can effectively communicate findings and make informed decisions based on survey results.

Central Tendency and Dispersion

Measures of Central Tendency

Top images from around the web for Measures of Central Tendency

• 

  summary statistics - Explaining Mean, Median, Mode in Layman's Terms - Cross Validated View original

  Is this image relevant?

• 

  Arithmetic mean - Wikipedia View original

  Is this image relevant?

• 

  Data Science for Water Professionals: Descriptive Statistics in R View original

  Is this image relevant?

• 

  summary statistics - Explaining Mean, Median, Mode in Layman's Terms - Cross Validated View original

  Is this image relevant?

• 

  Arithmetic mean - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Measures of Central Tendency

• 

  summary statistics - Explaining Mean, Median, Mode in Layman's Terms - Cross Validated View original

  Is this image relevant?

• 

  Arithmetic mean - Wikipedia View original

  Is this image relevant?

• 

  Data Science for Water Professionals: Descriptive Statistics in R View original

  Is this image relevant?

• 

  summary statistics - Explaining Mean, Median, Mode in Layman's Terms - Cross Validated View original

  Is this image relevant?

• 

  Arithmetic mean - Wikipedia View original

  Is this image relevant?

1 of 3

• Mean calculates the average value of a dataset by summing all values and dividing by the number of observations
• Median identifies the middle value in a sorted dataset, useful for datasets with extreme outliers
• Mode represents the most frequently occurring value in a dataset, particularly useful for categorical data
• Arithmetic mean computed by adding all values and dividing by the number of observations: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$
• Weighted mean assigns different weights to values based on their importance: $\bar{x}_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}$
• Geometric mean calculated by multiplying all values and taking the nth root: $\bar{x}_g = \sqrt[n]{x_1 \times x_2 \times ... \times x_n}$

Measures of Dispersion

• Standard deviation measures the average distance between each data point and the mean
• Variance represents the average squared deviation from the mean, calculated as the square of the standard deviation
• Range determines the difference between the maximum and minimum values in a dataset
• Interquartile range (IQR) measures the spread of the middle 50% of the data
• Population standard deviation formula: $\sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}}$
• Sample standard deviation formula: $s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}$
• Coefficient of variation (CV) expresses standard deviation as a percentage of the mean: $CV = \frac{s}{\bar{x}} \times 100\%$

Data Visualization

Frequency Distributions and Histograms

• Frequency distributions organize data into categories or intervals, showing how often each value occurs
• Relative frequency distribution displays the proportion of observations in each category
• Cumulative frequency distribution shows the accumulated frequency up to each category
• Histograms represent frequency distributions graphically using adjacent rectangles
• Bin width in histograms affects the visual representation of data distribution
• Sturges' rule estimates the number of bins for a histogram: $k = 1 + 3.322 \log_{10}(n)$
• Frequency polygons connect the midpoints of histogram bars, useful for comparing multiple distributions

Box Plots and Outlier Detection

• Box plots (box-and-whisker plots) display five-number summaries of datasets
• Five-number summary includes minimum, first quartile (Q1), median, third quartile (Q3), and maximum
• Box in a box plot represents the interquartile range (IQR)
• Whiskers extend to the minimum and maximum values within 1.5 times the IQR
• Outliers plotted as individual points beyond the whiskers
• Modified box plots use different methods to identify outliers (Tukey's method)
• Side-by-side box plots facilitate comparison of multiple datasets or groups

Distribution Characteristics

Skewness and Asymmetry

• Skewness measures the asymmetry of a probability distribution
• Positive skew indicates a longer tail on the right side of the distribution
• Negative skew indicates a longer tail on the left side of the distribution
• Pearson's coefficient of skewness: $SK_p = \frac{3(\bar{x} - \text{median})}{s}$
• Fisher-Pearson standardized moment coefficient: $g_1 = \frac{m_3}{m_2^{3/2}}$, where $m_k$ is the kth central moment
• Bowley's coefficient of skewness: $SK_b = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1}$

Kurtosis and Tail Behavior

• Kurtosis measures the heaviness of the tails of a probability distribution
• Excess kurtosis compares the kurtosis of a distribution to that of a normal distribution
• Mesokurtic distributions have kurtosis similar to a normal distribution (excess kurtosis ≈ 0)
• Leptokurtic distributions have heavier tails and higher peaks (excess kurtosis > 0)
• Platykurtic distributions have lighter tails and flatter peaks (excess kurtosis < 0)
• Fisher's measure of kurtosis: $g_2 = \frac{m_4}{m_2^2} - 3$, where $m_k$ is the kth central moment
• Pearson's measure of kurtosis: $\beta_2 = \frac{m_4}{m_2^2}$