Statistics and probability

Statistics and probability

Statistics describes data, while probability describes uncertainty. Before calculating, identify what the observations represent and how they were collected. The mean, median and mode summarise a data set in different ways. A single summary cannot show every feature of a distribution, so it helps to inspect the values, their spread and any unusually large or small observations.

The arithmetic mean is the sum of the observations divided by the number of observations. The median is the middle value after arranging the observations in order. With an even number of values, average the two middle values. The mode is the most frequent value; a set can have more than one mode or no repeated value. Always order the values before locating a median.

The range is the maximum value minus the minimum value. It measures overall spread but depends only on the two extremes. If a new very large observation is added, both the range and the mean can change sharply. The median is often less affected by such an outlier. A good interpretation therefore states which measure was used and why that measure suits the question.

A frequency table records how many times each value occurs. To find its mean, multiply each value by its frequency, sum the products and divide by the total frequency. Do not divide by the number of rows. For grouped continuous data, class midpoints give an estimated mean because the individual observations within each class are unknown. Label an estimated result accordingly.

For equally likely outcomes, probability is the number of favourable outcomes divided by the number of possible outcomes. A fair six-sided die has six equally likely results, so the probability of an even result is. Probabilities lie between zero and one. The complement of an event has probability one minus the probability of that event.

For independent events, multiply their probabilities to find the probability that both occur. Drawing without replacement usually changes the composition of a collection, so independence cannot be assumed. For mutually exclusive events, add their probabilities to find the probability that either occurs. Read phrases such as ‘at least one’, ‘exactly one’ and ‘with replacement’ carefully, and use a tree diagram or a list of outcomes when necessary.

Choose a measure that answers the question

Statistics organises observations so that patterns can be described and compared. The mean uses every value, the median describes the middle of ordered data, and the mode identifies the most frequent value or category. An extreme value can strongly affect a mean while leaving a median almost unchanged. For incomes 20, 22, 24, 26 and 200 thousand naira, the mean is 58.4 thousand but the median is 24 thousand. Both are calculated correctly, yet they describe different aspects of this uneven distribution.

For a frequency table, multiply each value x by its frequency f, add the products, then divide by the total frequency: mean = Σfx/Σf. If scores 2, 3 and 5 occur 4, 2 and 4 times, the mean is (8 + 6 + 20)/10 = 3.4. For grouped intervals, use each class midpoint as a representative x. That gives an estimated mean, since the exact individual values within a class are unknown. State this limitation when the question distinguishes exact and estimated statistics.

Read charts according to their scales

A bar chart uses separate bars for categories, whereas a histogram represents numerical intervals with adjoining bars. If histogram intervals have unequal widths, bar height is frequency density, calculated as frequency/class width. The bar area then represents frequency. A class 10 units wide with frequency 20 has density 2; a class 5 units wide with frequency 15 has density 3. The taller second bar does not mean its frequency exceeds 20. Check the axis label instead of assuming every vertical scale counts people.

A pie chart allocates angle in proportion to frequency: category angle = category frequency/total × 360°. Cumulative frequency adds frequencies up to successive upper boundaries. Plotting cumulative frequency against upper class boundaries gives an ogive, from which estimates of median and quartiles can be read. For N observations, locate approximately N/2 for the median, N/4 for the lower quartile and 3N/4 for the upper quartile. Follow the conventions specified in a question when exact quartiles of a short ordered list are required.

Measure spread as well as centre

The range is highest minus lowest value. It is easy to calculate but depends on only two observations. Population variance is Σ(x − x̄)²/N, and population standard deviation is its square root. For 2, 4 and 6, the mean is 4, squared deviations are 4, 0 and 4, variance is 8/3 and standard deviation is √(8/3) ≈ 1.633. A sample variance estimate uses N − 1 instead, so read whether a population or a sample measure is requested. Never switch denominators without stating the convention.

Standard deviation has the original measurement unit; variance has its square. Two classes can have equal mean scores but different spreads, with one class more consistent than the other. Mean absolute deviation uses absolute deviations rather than squared deviations. For 2, 4 and 6 it is. Avoid adding raw signed deviations as a spread measure: deviations from the mean sum to zero even when the data are widely dispersed.

Build probabilities from an explicit experiment

A probability lies between 0 and 1 inclusive. For equally likely outcomes, probability is favourable outcomes divided by all possible outcomes. The equally likely condition matters: a biased coin need not give heads with probability 1/2. The complement rule is. For any events, P(A or and B). If the events are mutually exclusive, their intersection has probability zero. Mutually exclusive events cannot both happen in the same trial.

Independent events satisfy P(A and. Drawing with replacement from a well-mixed bag can preserve independence; drawing without replacement generally changes the second probability. From a bag containing 3 red and 2 blue counters, P(two red without. With replacement it is. A probability tree helps keep conditional denominators correct. Self-check: the median of 7, 2, 9, 4 is 5.5 after sorting. For a fair die, P(even or greater than because the qualifying outcomes are 2, 4, 5 and 6, counting 6 only once.

Count outcomes without treating categories as equally likely

Two fair dice have 36 equally likely ordered outcomes. Their possible sums 2 through 12 are not equally likely: sum 7 has six ordered outcomes, while sum 2 has only one. Thus. When counting choices of two distinct objects, decide whether order matters and whether replacement is allowed before constructing the sample space.

For grouped spread estimates, use class midpoints consistently. An estimated population variance can be written Σfx²/Σf − (Σfx/Σf)². The estimate reflects grouped information and should not be presented as an exact calculation from unavailable raw data. Retain enough digits in the mean to avoid a negative variance caused by premature rounding.

Estimate grouped location by interpolation

Use continuous intervals < 10, 1 < 20, 2 < 30 and 3 < 40, with frequencies 4, 8, 6 and 2. Total N = 20 and cumulative frequencies are 4, 12, 18 and 20. Plot the ogive at upper boundaries (10, 4), (20, 12), (30, 18), (40, 20), also including the starting boundary (0, 0). A cumulative count belongs at its upper boundary, not the class midpoint.

The median position is N/2 = 10, inside the 10-to-20 class. Assuming observations are spread uniformly within that class, interpolate as median ≈ L + [(N/2 − F)/f]h. Here L = 10, F = 4 is the cumulative frequency before the class, f = 8 is the class frequency, and h = 10 is its width. The result is 10 + [(10 − 4)/8]10 = 17.5. This is an estimate, not a recovered exact median of the unavailable individual measurements.

For the lower quartile use target cumulative count N/4 = 5, giving Q₁ ≈ 10 + [(5 − 4)/8]10 = 11.25. For the upper quartile use 3N/4 = 15, giving Q₃ ≈ 20 + [(15 − 12)/6]10 = 25. The interquartile range is 25 − 11.25 = 13.75 and the semi-interquartile range is half that. For a pth percentile, replace the target by pN/100 and locate its class first. Reading a drawn ogive gives the corresponding graphical estimate.

The modal class is the class containing the highest frequency for equal-width intervals. A simple grouped-mode estimate is L + [d₁/(d₁ + d₂)]h, where d₁ is modal frequency minus previous frequency and d₂ is modal frequency minus next frequency. For the same table, L = 10, d₁ = 8 − 4 = 4 and d₂ = 8 − 6 = 2, giving mode ≈ 16.67. With unequal widths compare frequency densities first; do not assume the highest raw frequency identifies the tallest histogram bar.

Compute grouped spread with the same representatives

Midpoints 5, 15, 25 and 35 give Σfx = 360, so the estimated mean is 18. Also Σfx² = 8,100. The estimated population variance is 8,100/20 − 18² = 81, with standard deviation 9. The estimated mean absolute deviation is [4|5 − 18| + 8|15 − 18| + 6|25 − 18| + 2|35 − 18|]/20 = 152/20 = 7.6. Use the same midpoint approximation and denominator convention throughout rather than mixing raw-data and grouped-data formulas.

Self-check with this table: the 80th-percentile target is 16, which lies in the 20-to-30 class, so its estimate is. Check that it exceeds the median and lies within its identified class. In an experiment giving 18 heads in 40 coin tosses, experimental probability of heads is 18/40 = 0.45. That relative frequency is observed data; it does not prove that a coin's long-run probability must equal 0.45. Distinguish the theoretical model from the finite experimental result.