Unit 7: Data Interpretation mind map

The most common family: percent more, percent less, one value as a percent of another, and percentage increase.

Reading a table for percentages

Percent more than and percent less than

The base is always the thing you compare WITH, the number after 'than'.

• Percent more = (bigger - smaller) / smaller x 100.
• Percent less = (bigger - smaller) / bigger x 100.
• Find both values first, then take the difference.
• Example: 66.67 percent more for profit 18 to 30.

Store A's profit rose from 18 to 30. The rise is 12, and 12/18 x 100 = 66.67 percent. If the same figures are asked as percent less, India 160 against Germany 250 gives 90/250 x 100 = 36 percent. The pair of figures is the same; the base changes.

One value as a percent of another

'What percent of' means value divided by the second value, times 100.

• 600 is what percent of 2500? 24 percent.
• 115 is what percent of 160? 71 7/8 percent.
• A total can be a percent of a total: 550 against 500 is 110 percent.
• A percent above 100 is normal when the first is larger.

Do not subtract anything. Find the two figures, divide, and multiply by 100. Convert each cell first if the table gives percentages of a total.

Percentage increase between years

Percentage increase uses the earlier year as the base.

• Increase = (later - earlier) / earlier x 100.
• Lenovo 20 thousand to 90 thousand: 70/20 x 100 = 350 percent.
• Town D 3,500 to 3,600: about 2.85 percent.
• Percentage of one year over another is later / earlier x 100.

Percent of one year over another differs from percentage increase by exactly 100. For 2021 against 2020 with totals 900 and 750, the first is 120 percent and the increase is 20 percent. Read the question wording.

Year of maximum percentage increase or share

Work out the percentage for every year, then pick the largest. Do not guess from raw numbers.

• Increase for each year: (this year - last year) / last year.
• Share for each year: item / total in that year.
• Near ties need one decimal place: 34.1 against 34.0.
• A smaller total can raise a share.

Production 35, 50, 70, 55, 75, 60, 85, 90 gives 2012 as 42.9 percent, higher than 2013 at 40 percent. The biggest absolute rise is not always the biggest percentage rise, because the base differs.

Working with ratios

Ratio of two values

Find both quantities from the table, then reduce the ratio.

• 720 : 160 reduces to 9 : 2.
• 300 : 396 reduces to 25 : 33.
• Divide by the largest common factor.
• Match the order asked: first to second.

Ratio questions often hide a derived value. A shop's speakers may come from the camera count and a given camera-to-speaker ratio, such as 10 : 7, so work that out first.

Ratio after growth

Apply each growth rate to its own base, then take the ratio.

• Raise by 60 percent: multiply by 1.6. By 20 percent: 1.2.
• IT 320 x 1.6 = 512, marketing 560 x 1.2 = 672. Ratio 16 : 21.
• If only a relation is given, let one quantity be E and carry it through.
• PSU income 2E x 1.17 against foreign expenditure E x 1.11 gives 78 : 37.

Do not add the percentages together. Each department or bank has its own multiplier.

Percent less within a ratio

When a ratio is given, percent less needs no actual numbers.

• Rural : urban = 6 : 7: rural is 1 part in 7 less, 14.29 percent.
• The base is the larger quantity.

Use parts, not people. The question still gives extra data, but you can ignore it if the ratio already answers it.

Females : males = 4 : 5 means females are 1 part out of 5 less. That is 20 percent.

Sums, differences, averages and counting questions that need a column or a row added up.

Totals and averages

Sums and differences

Add the right cells first, then take the difference.

• Magazines in 2022 total 130 and in 2020 total 80. Difference 50.
• College totals: 3,800 in 2020 against 3,100 in 2023. Difference 700.
• Aggregate marks: convert each percentage using the subject maximum, then add.
• Read 'together' as a sum.

Aggregate marks for a student are percentage times maximum per subject. Maths 60 percent of 100 is 60 and chemistry 50 percent of 150 is 75. Add all subjects to get 352.5 for the whole set.

Average of a column

Average equals the total divided by the number of items.

• Girls in college A over six years: 10,320 / 6 = 1,720.
• College B over four years: 3,400 / 4 = 850.
• Average expenditure needs expenditure itself: income / (1 + profit percent).
• Count the items before dividing.

When only income and profit percent are given, expenditure is not shown. Get it first. For example 2017: 120 / 1.075 is about 111.6. Then average over the six years.

Years above or below the average

Compute the average once, then count the years that beat it.

• Total 540 over 7 years gives 77.14. Years above: 3.
• Total 1,15,500 over 7 years gives 16,500. Years below: 4.
• Check equal-to cases carefully.

Do not recompute the average for each year. Write it once, then scan the row.

Total 520 over 8 years gives average 65. Years above: 70, 75, 85, 90 are 4 years.

Counting rows above a threshold

Derive the value for every row, compare each with the threshold, and count.

• Clothes spending above Rs 41,800: persons B and E only. F at 41,400 does not qualify.
• Laptops above 2000: stores C and E. Store A has exactly 2,000.
• Exactly equal does not count as more than.

These need the most computation. Work through every row. Strict 'more than' excludes the boundary value, and 'more than or equal' includes it.

Age group I is more than 4 lakh in cities C and D only. B has exactly 4.0, not more.

Counting against several conditions

Test each person against the rule, then count the ones that pass all.

• Seventy percent or more in at least four subjects: count subjects per student.
• First division needs 65 percent aggregate of 1,000, which is 650. Only one student reaches it.
• Total maximum comes from adding each subject's maximum.

Find the cut-off in marks, not in percent, when subjects have different maximums.

Tables with a twist

Find the total from a sub-share

When a part is given in numbers, divide by its share to get the total.

• Japan 24,000 at 30 percent gives total 80,000. UK 10 percent gives 8,000.
• A difference of shares: 30 percent of total is 1,65,000, so total 5,50,000.

First turn every given number into a share of the total. Then scale up and read the answer from the second table.

Switzerland is 25 percent of Others at 20 percent, so it is 5 percent. 25 lakh gives total 500 lakh.

Others are 20 percent and equal 5 lakh, so total 25 lakh and USA at 40 percent is 10 lakh.

Two tables, one question

Question parts live in different tables. Link them through a shared total.

• Take total as 100 if no actual number is needed.
• USA 40 and below-40 tourists 30 gives 4 : 3.
• Derive share times state total, then compare.

Percent tables do not show people. Multiply the percent by the right state or city total first.

Insurance in Rajasthan 18.5 percent of 3,120 against construction in Haryana 14.4 percent of 3,880: about 103 percent.

Hidden totals from a percent table

A table in percent needs the total or a given figure before you find numbers.

• When both use the same total, 23/20 = 1.15 directly.
• Chess plus squash is 65 percent when carrom is 35 percent.
• Total = given number divided by its share.

Where the base cancels, skip actual numbers and compare the shares. Where the totals differ, compute the numbers.

Boys from city A are 23 percent of 24,000 = 5,520. City B is 20 percent = 4,800. A is 15 percent more.

Cumulative percentage tables

In a cumulative table, a band is the difference between two cumulative figures.

• Subtract the other subject's band to get this subject's band.
• Hindi 130 in the band, so Punjabi = 300 - 130 = 170.
• Ratio of bands: 130 : 50 = 13 : 5.

Work the combined band first, then subtract one subject. Cumulative figures are totals up to a mark, not marks within one band.

1,200 students in all. Cumulative 65 percent = 780 and 90 percent = 1,080. Band 50-59 = 300 in all.

Mean, median, mode, geometric mean, and the measures of dispersion: standard deviation and coefficient of variation.

Central tendency

Mean, median and mode

Mean is the sum over the count. Median is the middle value. Mode is the most frequent.

• Mean of 7, 5, 9, 8, 15, 3, 8, 9 is 64/8 = 8.
• Mode of 2, 3, 7, 1, 3, 2, 3 is 3.
• Mean of x, x+2, x+4, x+6, x+8 equal to 11 gives x = 7.

Always sort before finding the median. In Excel, AVERAGE returns the mean, COUNT the number of entries, MEDIAN the middle value and MODE the most frequent.

Median of 4, 4, 5, 7, 6, 7, 7, 12, 3: sort first. Nine values, the 5th is 6.

Mean after a transformation

Use the algebra of means. Express the new mean through the old sum.

• Mean of x/y, 1, z/y equals K/y.
• Mean of a x_i together with x_i / a is (1/2)(a + 1/a) times the mean.
• Adding or subtracting a constant shifts the mean by that constant.

Write the sum from the mean, substitute, and simplify. Count how many values there are in the new data set before dividing.

If the mean of x, y, z is K, then x + y + z = 3K.

Geometric mean and the median from a table

Geometric mean is the nth root of the product. For a grouped table, use the median formula.

• 9, 12 and 16: product 1728, cube root 12.
• Median = L + ((N/2 - cf)/f) x h.
• Median 14.4 lies in class 12-18, so L = 12 and h = 6.
• Total frequency 20 gives x + y = 10. Answer x = 4, y = 6.

Locate the median class first from N/2. Then put the known values into the formula and solve for the missing frequency.

Measures of spread

Which measures show spread

Range, mean deviation and standard deviation measure spread. Mean, median and mode measure the centre.

• Dispersion: range, mean deviation, standard deviation, variance.
• Central tendency: mean, median, mode.
• A quartile is a position value, not a measure of spread.
• Quartile deviation does measure spread.

The option list mixes both groups to catch you. Tick only spread measures.

Coefficient of variation

CV is standard deviation divided by the mean. Compare spread relative to size.

• A: 12/50 = 0.24. B: 10/55 = 0.182. C: 14/60 = 0.233. D: 15/70 = 0.214.
• Increasing order: B, D, C, A.
• Multiply by 100 for percent; the order stays the same.
• Divide each SD by its own mean.

This came twice on the same day with new numbers. Work each pair, write the decimals, and sort.

Standard deviation from the mean of squares

Variance equals the mean of squares minus the square of the mean.

• sigma squared = (1/N) sum x_i squared - mu squared.
• mu 5, mean of squares 50: 50 - 25 = 25, sigma 5.
• mu 4 and 52 gives sigma 6. mu 3 and 58 gives 7.
• Take the square root last.

A second paper used mean 3 with mean of squares 90, giving sigma 9, then 10, 11 and 12. The pattern repeats with fresh numbers.

Deviation and the effect of a constant

Deviation is the gap from the mean. Subtracting a constant moves the mean but not the spread.

• Monthly expenses total 252 over 12 months. Mean 21.
• Least deviation of 1 is in October and November.
• Mean 47 and SD 9, subtract 3: new mean 44, SD still 9.
• Multiplying every value does change the SD.

Spread depends on the gaps between values. Shifting all values by the same amount keeps every gap.

Percentiles, deciles and quartiles in order, skewness and kurtosis, and which chart suits which job.

Position and shape

Percentiles, deciles and quartiles

Turn every measure into a percentile, then sort.

• Decile n is the 10n-th percentile.
• Quartile 1, 2, 3 are the 25th, 50th and 75th percentiles.
• Median is the 50th percentile.
• Sixth decile 60, 67th percentile 67, third quartile 75.

Median, sixth decile, 67th percentile, third quartile sort to 50, 60, 67, 75. In another paper the first quartile, third decile, 37th percentile and median sort to 25, 30, 37, 50.

Skewness and kurtosis

Skewness is lopsidedness. Kurtosis is the sharpness of the peak.

• Karl Pearson skewness = (mean - mode) / SD.
• SD from CV: SD = CV x mean / 100.
• A sharp peak is leptokurtic; a flat top is platykurtic.
• Sharpness of the peak is kurtosis.

For four distributions with given mean, mode and CV, first find SD from CV, then the skewness, then arrange the four. The key was A, C, D, B.

Choosing a chart

Match the chart to the task

Each chart answers one kind of question.

• Pie chart: proportions of a whole.
• Line chart: trends over time.
• Scatter plot: association between two variables.
• Histogram: frequency distribution. Ogive: cumulative frequencies.

Bar charts compare separate categories. A box plot shows spread. The wrong options in these items are the other charts, so match the task word first.

Pictorial presentation

Pictorial presentation means charts and graphs, not tables.

• Pie charts, bar charts and line graphs are pictorial.
• Cumulative and non-cumulative frequency tables are tables of numbers.
• Answer: pie, bar and line.

A table is data in rows and columns. A chart draws the same data as a picture.

Normal curve areas, binomial variance, chi-square, regression, correlation, and a few named tests.

Distributions

Standard normal areas

Use the symmetric standard normal table. Half the area is on each side of zero.

• P(0 to 1) = 0.3413. P(0 to 2) = 0.4772.
• P(Z at least -1) = 0.5 + 0.3413 = 0.84.
• P(-2 to 1) = 0.4772 + 0.3413 = 0.82.
• P(Z up to 2) = 0.98. P(Z beyond 2) = 0.0228.

Split an interval at zero and add the two halves. Tails come from one minus the area to the left. In increasing order the four probabilities came out D, C, A, B.

Binomial variance and chi-square

Binomial variance is n times p times (1 - p). Chi-square has mean equal to its degrees of freedom.

• n 7 and p 0.3: 1.47. n 5 and p 0.4: 1.20.
• n 8 and p 0.5: 2.00. n 6 and p 0.6: 1.44.
• Chi-square variance is two times its degrees of freedom.
• Three times the degrees of freedom is NOT a property.

Chi-square is also a test statistic, and for large degrees of freedom it approaches the normal distribution.

Relationships and tests

R-squared and correlation

R-squared is 1 minus RSS over TSS. The correlation is covariance over the product of the SDs.

• RSS 5 and TSS 20: R-squared 0.75.
• RSS 12 and TSS 32: 0.625.
• Cov 32 with SDs 8 and 7: r = 0.571.
• Cov 39 with SDs 6 and 13: r = 0.50.

Both formulas need only division. Work each option, write the decimal, then match or sort.

Choosing a test

Know what each named test is for.

• Z-test: significance of means, large sample or known SD.
• Chi-squared test: association between categories and goodness of fit.
• ANOVA: analysis of variance, testing whether three or more means are equal.
• F-test compares variances.

ANOVA compares the variance between groups with the variance within groups.