ACCA Applied Knowledge · Management Accounting

Summarising and analysing data: formula sheet

Full chapter guide

Key formulas

Primary vs secondary data: Primary = collected first-hand for this purpose; Secondary = already collected by someone else; The test is who collected the data and why, not whether it is internal or external.
Quantitative vs qualitative data: Quantitative = numerical and measurable; Qualitative = descriptive, based on opinion or quality; Ratings scored from 1 to 5 are still based on opinion, but the scores are numerical.
Discrete vs continuous data: Discrete = counted, separate values; Continuous = measured, any value in a range; Ask: can the value sensibly be a fraction or decimal? If yes, it is continuous.
Internal vs external source: Internal = from within the organisation; External = from outside it; Internal data can be primary or secondary. Check the wording of the question.
Systematic sampling interval: Interval (n) = population size ÷ sample size; Pick a random start within the first n items, then take every nth item. Round sensibly if the answer is not whole.
Proportionate stratified sample: Sample from a stratum = (stratum size ÷ population size) × total sample size; Use this to allocate the sample across strata. Round so the total still equals the sample size.
Stratified vs cluster rule: Stratified: sample from every group. Cluster: sample all items from some groups.; Strata should differ from each other and be similar inside. Clusters should each resemble the whole population.
Class width: Class width = upper boundary − lower boundary; For 10 – under 20 the width is 10. Use boundaries, not stated limits, when classes have gaps.
Frequency density: Frequency density = frequency ÷ class width; Plot this on the vertical axis of a histogram when class widths are unequal.
Standard-width adjustment: Adjusted height = frequency ÷ (class width ÷ standard width); An alternative to frequency density. A class twice the standard width has its frequency halved.
Pie chart angle: Angle = (category value ÷ total) × 360°; Angles must add to 360°.
Cumulative frequency: Cumulative frequency = running total of frequencies; Plot against the upper class boundary for an ogive.
Relative frequency: Relative frequency = frequency ÷ total frequency; Can be shown as a fraction, decimal or percentage.
Median position on an ogive: Read the value at cumulative frequency = n ÷ 2; Quartiles are read at n ÷ 4 and 3n ÷ 4.
Arithmetic mean (raw data): x̄ = Σx ÷ n; Add all values, divide by the number of values.
Mean (frequency or grouped data): x̄ = Σfx ÷ Σf; For grouped data, x is the class mid-point. The result is an estimate.
Class mid-point: Mid-point = (lower limit + upper limit) ÷ 2; Check class boundaries are continuous before using this.
Median position (raw data, ordered): Position = (n + 1) ÷ 2; If the position is a half, take the average of the two middle values.
Median (grouped data): Median = L + [(n ÷ 2 − cf) ÷ f] × w; L = lower boundary of median class, cf = cumulative frequency before it, f = its frequency, w = class width.
Mode (grouped data): Mode = L + [d1 ÷ (d1 + d2)] × w; L = lower boundary of modal class, d1 = modal frequency minus previous frequency, d2 = modal frequency minus next frequency. Equal class widths assumed.
Weighted average: Weighted mean = Σwx ÷ Σw; w is the weight, such as units, hours or proportion.
Range: Range = highest value − lowest value; Uses only two values, so it is affected by extreme figures.
Interquartile range: IQR = Q3 − Q1; Measures the spread of the middle 50% of the data. Q1 and Q3 are the lower and upper quartiles.
Variance (population, ungrouped): σ² = Σ(x − x̄)² ÷ n; Use n when the data is treated as the whole population, as is usual in MA questions. Check whether the question says sample.
Variance (shortcut form): σ² = (Σx² ÷ n) − x̄²; Often faster. Mean of the squares minus the square of the mean.
Standard deviation: σ = √variance; In the same units as the data.
Variance for a frequency distribution: σ² = (Σfx² ÷ Σf) − x̄²; Here x̄ = Σfx ÷ Σf. Σf is the total frequency.
Coefficient of variation: CV = (standard deviation ÷ mean) × 100%; Compares relative spread. Lower CV means more consistent data.
Simple price index: Price index = (Current price ÷ Base price) × 100; Works the same for quantity or any other single series.
Laspeyres price index: Σ(P1 × Q0) ÷ Σ(P0 × Q0) × 100; P0 and P1 are base and current prices. Q0 is base-period quantity. Weights stay fixed.
Paasche price index: Σ(P1 × Q1) ÷ Σ(P0 × Q1) × 100; Q1 is current-period quantity. Weights change each period.
Laspeyres quantity index: Σ(Q1 × P0) ÷ Σ(Q0 × P0) × 100; Quantities change, prices held at base. Paasche quantity index uses current prices: Σ(Q1 × P1) ÷ Σ(Q0 × P1) × 100.
Deflating a value: Real value = Current value ÷ Price index × 100; Gives the value in base-period prices.
Rate of change between periods: % change = (Index later − Index earlier) ÷ Index earlier × 100; Do not simply subtract index points unless the earlier index is 100.
Changing the base: New index = Old index ÷ Old index of new base period × 100; The new base year then shows 100.
Additive model: Y = T + S + R; Seasonal variation is an absolute amount. Seasonal variations should sum to zero over a full cycle.
Multiplicative model: Y = T × S × R; Seasonal variation is a factor or percentage. Factors should average 1 (sum to the number of periods in the cycle).
Seasonal variation, additive: S = Actual - Trend; Calculate for each period, then average the figures for the same season across years.
Seasonal variation, multiplicative: S = Actual ÷ Trend; Average the ratios for the same season across years.
Moving average, odd number of periods: Sum of n values ÷ n; Placed against the middle period. Example: 7-day average for daily data.
Centred moving average, even number of periods: (Average 1 + Average 2) ÷ 2, or (½ first + middle values + ½ last) ÷ 4 for quarters; Places the trend against an actual period.
Adjusting seasonal variations (additive): Adjustment = Total of averages ÷ number of seasons; subtract it from each average; Makes the variations sum to zero.
Forecast, additive: Forecast = Projected trend + Seasonal variation; Project the trend first, using the average change per period.
Forecast, multiplicative: Forecast = Projected trend × Seasonal factor; Use the factor for the correct season.
Seasonally adjusted figure: Additive: Actual - S. Multiplicative: Actual ÷ S; Removes the seasonal effect so you can see the underlying trend.
Correlation coefficient: r = (nΣxy − ΣxΣy) ÷ √[(nΣx² − (Σx)²) × (nΣy² − (Σy)²)]; n is the number of pairs. The answer must lie between −1 and +1. If it does not, you have made an arithmetic error.
Coefficient of determination: r² = r × r; Gives the proportion (or percentage) of variation in y explained by x. Never negative.
Regression line: y = a + bx; y is the dependent variable, x the independent variable. In cost estimation, a is fixed cost and b is variable cost per unit.
Slope (b): b = (nΣxy − ΣxΣy) ÷ (nΣx² − (Σx)²); The numerator is the same as in the r formula. Calculate it once and reuse it.
Intercept (a): a = (Σy − bΣx) ÷ n, or a = ȳ − b x̄; Calculate b first. ȳ and x̄ are the mean values of y and x.
High-low variable cost: b = (cost at highest activity − cost at lowest activity) ÷ (highest activity − lowest activity); Then fixed cost = total cost at either point − (b × activity at that point). Use the highest and lowest activity levels, not the highest and lowest costs.

Quick revision

Primary data is collected first-hand for a purpose; secondary data already exists and was collected by someone else.
Random sampling gives each item a known chance of selection; quota and convenience sampling are non-random and can bias results.
Mean = Σx ÷ n; the median is the middle value of ordered data; the mode is the most frequent value.
The mean uses every value but is affected by extreme values; the median is not.
Range = highest value - lowest value; standard deviation is the square root of variance.
A larger standard deviation means more spread around the mean.
Index = (current value ÷ base value) × 100.
Time series: actual = trend + seasonal variation + random variation in the additive model.
Forecasts become less reliable the further ahead you project the trend.
Correlation coefficient r lies from -1 to +1; it shows association, not cause and effect.
Regression line: y = a + bx, where b is the change in y for each one-unit change in x.
Linear programming finds the best use of scarce resources subject to constraints.

Common mistakes

Treating internal data as always primary and external data as always secondary. Fix: Use two separate tests. Primary or secondary depends on who collected the data and why. Internal or external depends on where the source sits. Last year's sales ledger used for a new forecast is internal but secondary.
Calling a measured quantity such as time or weight discrete. Fix: Ask how the value arises. If it is measured on a scale, it is continuous, even if reported to the nearest whole number.
Confusing stratified and cluster sampling. Fix: Stratified takes some items from every group. Cluster takes every item from some groups.
Calling quota sampling a random method. Fix: In quota sampling the interviewer picks people by judgement. Stratified picks randomly within each group.
Plotting raw frequency on a histogram with unequal class widths. Fix: Remember area equals frequency. Divide each frequency by its class width and plot frequency density.
Plotting an ogive against the class midpoint or lower boundary. Fix: Cumulative frequency is reached only at the end of a class, so plot it against the upper class boundary.
Taking a simple average when a weighted average is needed. Fix: If items have different quantities or importance, multiply by weights and divide by total weights.
Finding the median without ordering the data. Fix: Always sort from smallest to largest first.
Giving the variance when the question asks for standard deviation, or the other way round. Fix: Underline the measure asked for. If it is standard deviation, finish by taking the square root.
Forgetting to square the mean in the shortcut formula. Fix: Write the formula as (Σx² ÷ n) − (x̄)² and calculate x̄² as a separate line.

Exam tips

Read the question stem twice. Many marks are lost on one clue word such as 'published' or 'own survey'.
In multiple response questions, select exactly the number of answers stated. Extra choices usually score nothing.
Keep the two tests apart: primary or secondary is about who collected the data; internal or external is about where the source is.
For advantages and disadvantages, think cost, speed, relevance and reliability.
Do not spend long on these questions. They are quick marks if you know the definitions.
Learn one-line definitions for all six methods. Many objective questions are pure recognition.
In multiple response questions, read how many answers to select, and check each statement against the method.
For 'which method is most suitable' questions, match to the scenario constraint such as cost, spread or need for representation.