Frequency table maker and descriptive statistics calculator
Paste your data to build relative and cumulative frequency tables, descriptive measures and a histogram. Download the results as CSV, then practise with 120 lessons across four levels. Your data and progress stay on this device.
Frequency table, standard deviation and histogram
Calculation method, frequency table and histogram rules
The frequency table groups equal exact values, sorts them in ascending order and reports absolute, relative and cumulative frequencies. Quartiles use linear interpolation at (n−1)p. Histogram bins are [left, right), except the final inclusive bin. Identical values use a single bin. Automatic bin count: rounded-up √n, at most 10. Results and exports use the same data and settings.
How do I use this page?
- Read a lesson and its worked example.
- Enter an answer, check it and compare the solution steps.
- Generate another exercise or take the 10-question test.
What this page does not do: predict future lottery outcomes.
Course: 120 lessons
- 1. Data and sample size
- 2. Sum and units
- 3. Arithmetic mean
- 4. Median
- 5. Minimum, maximum and range
- 6. Even-sample median
- 7. Mode
- 8. Adding a constant
- 9. Changing units
- 10. Adding an observation
- 11. Removing an observation
- 12. Combining groups
- 13. Relative frequency
- 14. Bin midpoint
- 15. Percent versus percentage point
- 16. Relative frequency
- 17. Count from proportion
- 18. Missing-data share
- 19. Count after filtering
- 20. Total frequency
- 21. Minimum after sorting
- 22. Maximum after sorting
- 23. Second order statistic
- 24. Median rank
- 25. Distinct value count
- 26. Total from mean
- 27. Missing observation
- 28. Mean after replication
- 29. Deviation balance
- 30. Correction to a total
- 31. Interpolated quartiles
- 32. Interquartile range
- 33. Population and sample variance
- 34. Standard deviation
- 35. Outliers and histograms
- 36. Weighted mean
- 37. Deviation from the mean
- 38. SD after a shift
- 39. Mean absolute deviation
- 40. Cumulative count
- 41. Q1 position
- 42. Q3 position
- 43. Quarter interpolation
- 44. Zeroth percentile
- 45. Hundredth percentile
- 46. Zero weight
- 47. Normalising weights
- 48. Mean from frequency table
- 49. Scaling weights
- 50. Combining counts
- 51. Bin width
- 52. Density height
- 53. Bar area
- 54. Half-open boundary
- 55. Total area
- 56. Median with an extreme
- 57. Mean change after correction
- 58. Trimmed mean
- 59. Median absolute deviation
- 60. Robust location shift
- 61. Standard score
- 62. Sample variance
- 63. Variance after scaling
- 64. Lower Tukey fence
- 65. Range and scale
- 66. Value from z
- 67. Mean from z
- 68. SD from z
- 69. Negative z score
- 70. Units and z
- 71. SE from known SD
- 72. Fourfold sample size
- 73. Size for target SE
- 74. Variance of the mean
- 75. Dependent measurements
- 76. r from covariance
- 77. Covariance from r
- 78. Reversing an axis
- 79. Shifting an axis
- 80. Deterministic relation
- 81. Line prediction
- 82. Positive residual
- 83. Squared residual
- 84. Mean squared error
- 85. R squared
- 86. CV percent
- 87. SD from CV
- 88. Mean from CV
- 89. Mass unit conversion
- 90. Adding a constant and CV
- 91. Duplicate IDs
- 92. Zeros replacing missingness
- 93. Mixed units
- 94. Complete pairs
- 95. Sample coverage
- 96. Margin of error
- 97. Lower confidence bound
- 98. Upper confidence bound
- 99. Interval width
- 100. Repeated procedure
- 101. Relative mean change
- 102. Difference in shares
- 103. Rate per thousand
- 104. Pooled group mean
- 105. Standardised difference
- 106. Validation size
- 107. Absolute error
- 108. Root mean square error
- 109. Matched-pair count
- 110. Train–test gap
- 111. Project IQR
- 112. Project upper fence
- 113. Project lower fence
- 114. Observation on a fence
- 115. Whisker versus fence
- 116. Mean versus median in a report
- 117. Exclusion share
- 118. Display precision
- 119. No dispersion
- 120. Comparing SD conventions
Enter a fraction such as 1/3 or round as instructed beside the question (three or eight decimal places).
Optional local storage only. No account or server submission. Turn off to remove saved progress.
1. Data and sample size
Level 1
An observation is one measured value. Count all observations, including repeated values. First check units and missing data.
Worked example
Data 2; 2; 8 have n = 3 despite only two distinct values.
Watch out
A missing measurement is not zero. Do not replace unknown values with zero.
2. Sum and units
Level 1
The sum adds all observations and retains their unit. Only add comparable quantities.
Worked example
2 kg + 3 kg + 5 kg = 10 kg.
Watch out
Convert grams to kilograms before addition. A temperature sum is not an amount of heat.
3. Arithmetic mean
Level 1
The mean is sum divided by count. It uses every value and is therefore sensitive to extremes.
Worked example
For 2; 4; 9 the mean is 15/3 = 5.
Watch out
A mean of group means needs weights unless group sizes are equal.
4. Median
Level 1
Sort the data. For an odd count take the middle value; for an even count average the two middle values.
Worked example
1; 3; 8; 10: median = (3 + 8)/2 = 5.5.
Watch out
Sort before finding the middle. Median and mean are different.
5. Minimum, maximum and range
Level 1
Range is maximum minus minimum. It describes the full width but uses only two values.
Worked example
Data 3; 4; 12: range = 12 − 3 = 9.
Watch out
Two datasets with the same range can have very different distributions.
6. Even-sample median
Level 1
Sort the data and average the two middle values.
Worked example
2,4,8,10: median (4+8)/2=6.
Watch out
Do not choose just one middle value.
7. Mode
Level 1
The mode occurs most often; there can be more than one.
Worked example
2,2,3,5: mode 2.
Watch out
The mode need not equal the mean or median.
8. Adding a constant
Level 1
Adding c to every observation raises the mean by c.
Worked example
Mean 5 becomes 8 after adding 3 to every value.
Watch out
The number of observations is unchanged.
9. Changing units
Level 1
Multiplying data by c multiplies the mean by c.
Worked example
A mean of 2 metres is 200 centimetres.
Watch out
State the result unit.
10. Adding an observation
Level 1
Recover the sum as n × mean, add the new value and divide by n+1.
Worked example
4 values averaging 5 plus a new 10: (20+10)/5=6.
Watch out
Do not average means without weights.
11. Removing an observation
Level 1
Subtract the removed value from the sum and reduce count by 1.
Worked example
5 values, sum 30, remove 10: 20/4=5.
Watch out
Removing the only observation leaves an empty dataset.
12. Combining groups
Level 1
Compute the combined mean from group sums and total count.
Worked example
2 people average 3, 4 average 6: 30/6=5.
Watch out
Different group sizes require different weights.
13. Relative frequency
Level 1
Divide the class count by the total sample count.
Worked example
8 observations out of 20 is 40%.
Watch out
Do not divide by the number of histogram bins.
14. Bin midpoint
Level 1
The midpoint of [a,b] is (a+b)/2.
Worked example
The 10–20 interval has midpoint 15.
Watch out
This is the midpoint of bounds, not necessarily the mean within the bin.
15. Percent versus percentage point
Level 1
Subtract percentage shares to obtain percentage points.
Worked example
A rise from 20% to 30% is 10 pp, or 50% relative.
Watch out
Always label the kind of change.
16. Relative frequency
Level 1 · Data and frequencies
Relative frequency is count divided by sample size. Missing values are not zeros.
Worked example
4 of 20 observations: proportion from 0 to 1? 4/20 = 0.2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
17. Count from proportion
Level 1 · Data and frequencies
Relative frequency is count divided by sample size. Missing values are not zeros.
Worked example
A dataset of 200 has 4%. How many observations? 200 × 4/100 = 8
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
18. Missing-data share
Level 1 · Data and frequencies
Relative frequency is count divided by sample size. Missing values are not zeros.
Worked example
Of 40 measurements, 4 are missing. Missing percentage? 4/40 × 100 = 10
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
19. Count after filtering
Level 1 · Data and frequencies
Relative frequency is count divided by sample size. Missing values are not zeros.
Worked example
From 40 rows, 3 were excluded. Remaining count? 40 − 3 = 37
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
20. Total frequency
Level 1 · Data and frequencies
Relative frequency is count divided by sample size. Missing values are not zeros.
Worked example
Disjoint categories have counts 4, 3, 5. Total size? 4 + 3 + 5 = 12
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
21. Minimum after sorting
Level 1 · Order statistics
Sort observations first. Rank is a position; a value is the observation at that position.
Worked example
Data 12, 4, 7. Minimum? min = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
22. Maximum after sorting
Level 1 · Order statistics
Sort observations first. Rank is a position; a value is the observation at that position.
Worked example
Data 12, 4, 7. Maximum? max = 12
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
23. Second order statistic
Level 1 · Order statistics
Sort observations first. Rank is a position; a value is the observation at that position.
Worked example
Data 12, 4, 7. Second smallest? sort → 4, 7, 12 = 7
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
24. Median rank
Level 1 · Order statistics
Sort observations first. Rank is a position; a value is the observation at that position.
Worked example
There are 11 sorted observations. Median position (1-based)? (n + 1)/2 = 6
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
25. Distinct value count
Level 1 · Order statistics
Sort observations first. Rank is a position; a value is the observation at that position.
Worked example
Data 4, 4, 5, 6, 6. Distinct count? 3 = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
26. Total from mean
Level 1 · Mean and total
Total = count × mean. Combine groups using their sizes.
Worked example
5 measurements, mean 4. Total? 5 × 4 = 20
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
27. Missing observation
Level 1 · Mean and total
Total = count × mean. Combine groups using their sizes.
Worked example
Three values average 7. Two are 4 and 3. Third? 3 × 7 − 4 − 3 = 14
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
28. Mean after replication
Level 1 · Mean and total
Total = count × mean. Combine groups using their sizes.
Worked example
Mean 4. Every row is duplicated once. New mean? 2 × sum / (2 × n) = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
29. Deviation balance
Level 1 · Mean and total
Total = count × mean. Combine groups using their sizes.
Worked example
The arithmetic mean is 4. Sum of signed deviations from that mean? Σ(x − mean) = 0 = 0
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
30. Correction to a total
Level 1 · Mean and total
Total = count × mean. Combine groups using their sizes.
Worked example
Total 80. Entered 6 instead of 3. Correct total? 80 − 3 = 77
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
31. Interpolated quartiles
Level 2
Use h = (n−1)p in the sorted zero-indexed list. The fractional part interpolates between neighbours. Q1 uses p = 0.25 and Q3 uses 0.75.
Worked example
For 0; 10; 20; 30: Q1 = 7.5, Q3 = 22.5.
Watch out
Software uses different quartile conventions. Match the method before comparing results.
32. Interquartile range
Level 2
IQR = Q3 − Q1 describes the spread of the middle portion. It is less sensitive to isolated extremes than the full range.
Worked example
Q1 = 4 and Q3 = 12 give IQR = 8.
Watch out
IQR is not a percentage. It retains the original measurement unit.
33. Population and sample variance
Level 2
Subtract the mean, square each difference and sum. Divide by n for the described population, or usually by n−1 to estimate variance from a sample.
Worked example
For 2; 4; 6 squared deviations total 8. Population variance = 8/3; sample variance = 4.
Watch out
For one observation the n−1 estimator is undefined; the calculator shows no result.
34. Standard deviation
Level 2
Standard deviation is the square root of variance. It returns to the original unit and describes spread around the mean.
Worked example
Variance 9 cm² gives standard deviation 3 cm.
Watch out
Normal-distribution percentage rules do not apply to every dataset.
35. Outliers and histograms
Level 2
Tukey fences are Q1 − 1.5 IQR and Q3 + 1.5 IQR. Investigate values beyond them. Histograms count observations in bins; bin width changes their appearance.
Worked example
Q1 = 2, Q3 = 6: IQR = 4 and upper fence = 12. Value 13 is flagged.
Watch out
An outlier is not automatically an error. Check its source before removing it.
36. Weighted mean
Level 2
Divide the sum of value-weight products by total positive weight.
Worked example
Value 2 with weight 1 and 8 with weight 2: (2+16)/3=6.
Watch out
Do not divide by the number of distinct values.
37. Deviation from the mean
Level 2
Signed deviation is x−mean; its sign shows direction.
Worked example
x=3, mean=5: deviation −2.
Watch out
A deviation is not a standard deviation.
38. SD after a shift
Level 2
Adding the same constant changes neither spread nor SD.
Worked example
1,3,5 have the same SD as 11,13,15.
Watch out
The mean changes even though SD does not.
39. Mean absolute deviation
Level 2
Average absolute deviations from the mean.
Worked example
2,4,6: (2+0+2)/3=4/3.
Watch out
Do not confuse this with median absolute deviation.
40. Cumulative count
Level 2
Add counts across classes up to the stated boundary.
Worked example
Bin counts are 2,3,4: through the second there are 5.
Watch out
Define whether the boundary includes equality.
41. Q1 position
Level 2 · R7 quartiles
R7 interpolates at zero-based h=(N−1)p. Do not mix quartile conventions.
Worked example
N=5. Zero-based Q1 position? (N−1)/4 = 1
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
42. Q3 position
Level 2 · R7 quartiles
R7 interpolates at zero-based h=(N−1)p. Do not mix quartile conventions.
Worked example
N=5. Zero-based Q3 position? .75 × (N−1) = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
43. Quarter interpolation
Level 2 · R7 quartiles
R7 interpolates at zero-based h=(N−1)p. Do not mix quartile conventions.
Worked example
Between 4 and 16, interpolate with weight 0.25. Result? 4 + .25 × 12 = 7
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
44. Zeroth percentile
Level 2 · R7 quartiles
R7 interpolates at zero-based h=(N−1)p. Do not mix quartile conventions.
Worked example
Data 4, 7, 10. Percentile 0? P0 = minimum = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
45. Hundredth percentile
Level 2 · R7 quartiles
R7 interpolates at zero-based h=(N−1)p. Do not mix quartile conventions.
Worked example
Data 4, 7, 10. Percentile 100? P100 = maximum = 10
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
46. Zero weight
Level 2 · Weights and groups
Weights are non-negative with a positive total. Weighted mean = Σwx/Σw.
Worked example
Values 4 and 1000 with weights 1 and 0. Weighted mean? 4 × 1 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
47. Normalising weights
Level 2 · Weights and groups
Weights are non-negative with a positive total. Weighted mean = Σwx/Σw.
Worked example
Weights 4 and 3. Normalised first weight? 4/(4+3) = 0.5714285714
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
48. Mean from frequency table
Level 2 · Weights and groups
Weights are non-negative with a positive total. Weighted mean = Σwx/Σw.
Worked example
Value 2 occurs 4 times, 5 occurs 3 times. Mean? (2×4+5×3)/(4+3) = 3.285714286
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
49. Scaling weights
Level 2 · Weights and groups
Weights are non-negative with a positive total. Weighted mean = Σwx/Σw.
Worked example
Weighted mean 4. Multiply every weight by 3. New mean? Σ(cw)x / Σ(cw) = Σwx / Σw = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
50. Combining counts
Level 2 · Weights and groups
Weights are non-negative with a positive total. Weighted mean = Σwx/Σw.
Worked example
Groups contain 4 and 3 measurements. Denominator of combined mean? 4 + 3 = 7
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
51. Bin width
Level 2 · Histogram and density
A density bar area is a proportion. With unequal widths, height is not count.
Worked example
Interval [4, 7). Width? 7 − 4 = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
52. Density height
Level 2 · Histogram and density
A density bar area is a proportion. With unequal widths, height is not count.
Worked example
A bin of width 3 contains 4 of 20 observations. Density? 4/(20×3) = 0.06666666667
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
53. Bar area
Level 2 · Histogram and density
A density bar area is a proportion. With unequal widths, height is not count.
Worked example
Density 0.02, width 3. Area? .02 × 3 = 0.06
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
54. Half-open boundary
Level 2 · Histogram and density
A density bar area is a proportion. With unequal widths, height is not count.
Worked example
Bins [0,10), [10,20]. Which contains 10? Enter 1 or 2. 10 ∉ [0,10), 10 ∈ [10,20] = 2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
55. Total area
Level 2 · Histogram and density
A density bar area is a proportion. With unequal widths, height is not count.
Worked example
A density histogram includes all observations. Total bar area? Σ count/N = 1 = 1
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
56. Median with an extreme
Level 2 · Robust summaries
Median depends on ranks; mean uses every value. An extreme value affects measures differently.
Worked example
Data 4, 5, 1000. Median? middle ordered value = 5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
57. Mean change after correction
Level 2 · Robust summaries
Median depends on ranks; mean uses every value. An extreme value affects measures differently.
Worked example
One of 5 values increased by 15. Increase in mean? 15/5 = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
58. Trimmed mean
Level 2 · Robust summaries
Median depends on ranks; mean uses every value. An extreme value affects measures differently.
Worked example
Data 0, 4, 7, 10, 1000. Trim one value at each end. Mean? (4+7+10)/3 = 7
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
59. Median absolute deviation
Level 2 · Robust summaries
Median depends on ranks; mean uses every value. An extreme value affects measures differently.
Worked example
Data 4, 7, 10. Median absolute deviation from median? median(3,0,3) = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
60. Robust location shift
Level 2 · Robust summaries
Median depends on ranks; mean uses every value. An extreme value affects measures differently.
Worked example
Median 4. Add 3 to each value. New median? 4 + 3 = 7
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
61. Standard score
Level 3
For positive standard deviation, z=(x−mean)/SD.
Worked example
x=14, mean=10, SD=2: z=2.
Watch out
The score is undefined when SD=0.
62. Sample variance
Level 3
Divide squared deviations by n−1 for the sample estimator.
Worked example
Data 2,4,6: squared deviations sum to 8, sample variance 8/2=4.
Watch out
Name the measure before choosing n or n−1.
63. Variance after scaling
Level 3
Multiplying all data by c multiplies variance by c².
Worked example
Variance 3 under scale 2 becomes 12.
Watch out
Variance has squared units.
64. Lower Tukey fence
Level 3
The lower fence is Q1−1.5 × IQR. Values below it deserve review.
Worked example
Q1=4 and IQR=2: fence 1.
Watch out
Crossing the fence does not prove a data error.
65. Range and scale
Level 3
For positive scale c, the range is multiplied by c.
Worked example
Range 5 under scale 3 gives 15.
Watch out
Adding a constant does not affect range.
66. Value from z
Level 3 · Inverse standardisation
z=(x−μ)/σ, hence x=μ+zσ. A z score does not establish normality.
Worked example
μ=4, σ=3, z=2. x? 4 + 2×3 = 10
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
67. Mean from z
Level 3 · Inverse standardisation
z=(x−μ)/σ, hence x=μ+zσ. A z score does not establish normality.
Worked example
x=10, σ=3, z=2. μ? x − zσ = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
68. SD from z
Level 3 · Inverse standardisation
z=(x−μ)/σ, hence x=μ+zσ. A z score does not establish normality.
Worked example
x=10, μ=4, z=2. σ? (x−μ)/z = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
69. Negative z score
Level 3 · Inverse standardisation
z=(x−μ)/σ, hence x=μ+zσ. A z score does not establish normality.
Worked example
x=1, μ=4, σ=3. z? (x−μ)/σ = -1
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
70. Units and z
Level 3 · Inverse standardisation
z=(x−μ)/σ, hence x=μ+zσ. A z score does not establish normality.
Worked example
z=2 in metres. Convert all data and parameters to centimetres. New z? 100(x−μ)/(100σ) = 2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
71. SE from known SD
Level 3 · Standard error
For independent observations with equal variance, mean SE = σ/√n. SE is not data SD.
Worked example
σ=8, n=4. Mean SE? 8/√4 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
72. Fourfold sample size
Level 3 · Standard error
For independent observations with equal variance, mean SE = σ/√n. SE is not data SD.
Worked example
SE=4. Sample size increases fourfold at fixed σ. New SE? 4/2 = 2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
73. Size for target SE
Level 3 · Standard error
For independent observations with equal variance, mean SE = σ/√n. SE is not data SD.
Worked example
σ=4, target SE=2. Minimum n? n=(σ/SE)² = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
74. Variance of the mean
Level 3 · Standard error
For independent observations with equal variance, mean SE = σ/√n. SE is not data SD.
Worked example
Observation variance 20, n=5 independent observations. Variance of mean? 20/5 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
75. Dependent measurements
Level 3 · Standard error
For independent observations with equal variance, mean SE = σ/√n. SE is not data SD.
Worked example
Two observations each have variance 4 and covariance 2. Variance of their mean? (4+4+2×2)/4 = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
76. r from covariance
Level 3 · Covariance and correlation
r=cov/(SDx SDy). Correlation measures linear association, not causation.
Worked example
cov=6, SDx=4, SDy=3. r? cov/(SDx SDy) = 0.5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
77. Covariance from r
Level 3 · Covariance and correlation
r=cov/(SDx SDy). Correlation measures linear association, not causation.
Worked example
r=0.5, SDx=4, SDy=3. cov? .5 × 4 × 3 = 6
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
78. Reversing an axis
Level 3 · Covariance and correlation
r=cov/(SDx SDy). Correlation measures linear association, not causation.
Worked example
r=0.6. Replace X with −X. New r? r(−X,Y)=−r(X,Y) = -0.6
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
79. Shifting an axis
Level 3 · Covariance and correlation
r=cov/(SDx SDy). Correlation measures linear association, not causation.
Worked example
r=0.6. Add 4 to X. New r? r(X+c,Y)=r(X,Y) = 0.6
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
80. Deterministic relation
Level 3 · Covariance and correlation
r=cov/(SDx SDy). Correlation measures linear association, not causation.
Worked example
X varies, Y=4X+3. Pearson correlation? positive linear relation → r=1 = 1
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
81. Line prediction
Level 3 · Regression and residuals
Residual = observation − prediction. Training fit does not establish out-of-sample accuracy.
Worked example
ŷ=4x+3, x=3. ŷ? 4×3+3 = 15
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
82. Positive residual
Level 3 · Regression and residuals
Residual = observation − prediction. Training fit does not establish out-of-sample accuracy.
Worked example
y=7, ŷ=4. Residual? y−ŷ = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
83. Squared residual
Level 3 · Regression and residuals
Residual = observation − prediction. Training fit does not establish out-of-sample accuracy.
Worked example
Residual = −4. Squared residual? (−4)² = 16
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
84. Mean squared error
Level 3 · Regression and residuals
Residual = observation − prediction. Training fit does not establish out-of-sample accuracy.
Worked example
Residuals 4, −4. Mean squared error? (4²+(-4)²)/2 = 16
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
85. R squared
Level 3 · Regression and residuals
Residual = observation − prediction. Training fit does not establish out-of-sample accuracy.
Worked example
In simple linear regression with intercept, r=0.8. R²? .8² = 0.64
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
86. CV percent
Level 3 · Relative variability
CV=100 SD/mean requires a meaningful zero and positive mean. Do not use for Celsius.
Worked example
Positive masses: mean 40, SD=4. CV in %? 4/40 × 100 = 10
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
87. SD from CV
Level 3 · Relative variability
CV=100 SD/mean requires a meaningful zero and positive mean. Do not use for Celsius.
Worked example
Mean 40, CV=20%. SD? .2 × 40 = 8
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
88. Mean from CV
Level 3 · Relative variability
CV=100 SD/mean requires a meaningful zero and positive mean. Do not use for Celsius.
Worked example
SD=4, CV=25%. Mean? 4/.25 = 16
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
89. Mass unit conversion
Level 3 · Relative variability
CV=100 SD/mean requires a meaningful zero and positive mean. Do not use for Celsius.
Worked example
CV=15% in kg. Convert all values to grams. CV in %? 1000 SD/(1000 mean) = 15
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
90. Adding a constant and CV
Level 3 · Relative variability
CV=100 SD/mean requires a meaningful zero and positive mean. Do not use for Celsius.
Worked example
Mean 4, SD=3. Add 4 to every value. New CV in %? 3/8 × 100 = 37.5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
91. Duplicate IDs
Level 4 · Data audit
Check units, missingness, duplicates and count before calculating. Corrections need justification.
Worked example
File has 7 rows and 4 distinct IDs. Excess rows? rows−unique IDs = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
92. Zeros replacing missingness
Level 4 · Data audit
Check units, missingness, duplicates and count before calculating. Corrections need justification.
Worked example
Two measurements: 10 and 20. A missing value was entered as 0. Reduction in mean? 15−10 = 5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
93. Mixed units
Level 4 · Data audit
Check units, missingness, duplicates and count before calculating. Corrections need justification.
Worked example
Masses: 4 kg and 300 g. Total kg? 4+300/1000 = 4.3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
94. Complete pairs
Level 4 · Data audit
Check units, missingness, duplicates and count before calculating. Corrections need justification.
Worked example
Of 40 pairs, 3 lack X or Y. Complete pairs? 40−3 = 37
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
95. Sample coverage
Level 4 · Data audit
Check units, missingness, duplicates and count before calculating. Corrections need justification.
Worked example
Checked 4 of 20 records. Percentage? 4/20×100 = 20
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
96. Margin of error
Level 4 · Confidence intervals
Under a normal model with known σ, a 95% mean interval has margin 1.96σ/√n. This is a frequentist procedure.
Worked example
SE=4. Normal 95% margin? 1.96 × 4 = 7.84
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
97. Lower confidence bound
Level 4 · Confidence intervals
Under a normal model with known σ, a 95% mean interval has margin 1.96σ/√n. This is a frequentist procedure.
Worked example
Mean 40, margin 3. Lower bound? 40−3 = 37
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
98. Upper confidence bound
Level 4 · Confidence intervals
Under a normal model with known σ, a 95% mean interval has margin 1.96σ/√n. This is a frequentist procedure.
Worked example
Mean 40, margin 3. Upper bound? 40+3 = 43
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
99. Interval width
Level 4 · Confidence intervals
Under a normal model with known σ, a 95% mean interval has margin 1.96σ/√n. This is a frequentist procedure.
Worked example
Margin 3. Full interval width? 2 × 3 = 6
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
100. Repeated procedure
Level 4 · Confidence intervals
Under a normal model with known σ, a 95% mean interval has margin 1.96σ/√n. This is a frequentist procedure.
Worked example
1,000 independent intervals with exact 95% coverage. Expected number covering the parameter? 1000 × .95 = 950
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
101. Relative mean change
Level 4 · Comparing groups
Distinguish absolute changes, relative changes and percentage points. Unequal groups require weights.
Worked example
Mean increased from 16 to 20. Percentage increase? (5a−4a)/4a×100 = 25
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
102. Difference in shares
Level 4 · Comparing groups
Distinguish absolute changes, relative changes and percentage points. Unequal groups require weights.
Worked example
Share 4%, then 7%. Difference in percentage points? new−old = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
103. Rate per thousand
Level 4 · Comparing groups
Distinguish absolute changes, relative changes and percentage points. Unequal groups require weights.
Worked example
4 events per 200 people. Rate per 1,000? 4/200×1000 = 20
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
104. Pooled group mean
Level 4 · Comparing groups
Distinguish absolute changes, relative changes and percentage points. Unequal groups require weights.
Worked example
10 people average 4, 30 people average 3. Combined mean? (10a+30b)/40 = 3.25
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
105. Standardised difference
Level 4 · Comparing groups
Distinguish absolute changes, relative changes and percentage points. Unequal groups require weights.
Worked example
Mean difference 8, chosen common SD=4. Difference divided by SD? 2a/a = 2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
106. Validation size
Level 4 · Model checks
Correlation is not causation. Evaluate predictions on data not used for fitting.
Worked example
100 observations, 40 for training. Validation count? 100−40 = 60
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
107. Absolute error
Level 4 · Model checks
Correlation is not causation. Evaluate predictions on data not used for fitting.
Worked example
Prediction 4, actual 7. Absolute error? |actual−prediction| = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
108. Root mean square error
Level 4 · Model checks
Correlation is not causation. Evaluate predictions on data not used for fitting.
Worked example
Errors 4, −4. RMSE? √mean(errors²) = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
109. Matched-pair count
Level 4 · Model checks
Correlation is not causation. Evaluate predictions on data not used for fitting.
Worked example
X has 5 values, Y has 6. After identifying the unmatched record, how many Y values must be removed? (n+1)−n = 1
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
110. Train–test gap
Level 4 · Model checks
Correlation is not causation. Evaluate predictions on data not used for fitting.
Worked example
Training MSE 4, test MSE 7. Gap? test−train = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
111. Project IQR
Level 4 · Project: box plot
Compute quartiles using the chosen method. Whiskers end at observations inside Tukey fences, not at the fences.
Worked example
Q1=4, Q3=10. IQR? Q3−Q1 = 6
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
112. Project upper fence
Level 4 · Project: box plot
Compute quartiles using the chosen method. Whiskers end at observations inside Tukey fences, not at the fences.
Worked example
Q1=4, Q3=10. Upper fence? Q3+1.5 IQR = 19
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
113. Project lower fence
Level 4 · Project: box plot
Compute quartiles using the chosen method. Whiskers end at observations inside Tukey fences, not at the fences.
Worked example
Q1=4, Q3=10. Lower fence? Q1−1.5 IQR = -5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
114. Observation on a fence
Level 4 · Project: box plot
Compute quartiles using the chosen method. Whiskers end at observations inside Tukey fences, not at the fences.
Worked example
An observation equals the upper Tukey fence. Is it outside? 1=yes, 0=no. outside means >, not ≥ = 0
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
115. Whisker versus fence
Level 4 · Project: box plot
Compute quartiles using the chosen method. Whiskers end at observations inside Tukey fences, not at the fences.
Worked example
Upper fence 20; observations below it: 2, 5, 12, 18; also 30. Upper whisker? largest observation ≤20 = 18
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
116. Mean versus median in a report
Level 4 · Project: a defensible report
Report method, count, units and limitations. Do not choose summaries merely to support a claim.
Worked example
Data 1,1,1,1,20. Mean minus median? (4+5a)/5−1 = 3.8
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
117. Exclusion share
Level 4 · Project: a defensible report
Report method, count, units and limitations. Do not choose summaries merely to support a claim.
Worked example
Excluded 4 of 100 records. What percentage should be reported? 4/100×100 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
118. Display precision
Level 4 · Project: a defensible report
Report method, count, units and limitations. Do not choose summaries merely to support a claim.
Worked example
Round 4.126 to 2 decimals. third decimal 6 → round up = 4.13
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
119. No dispersion
Level 4 · Project: a defensible report
Report method, count, units and limitations. Do not choose summaries merely to support a claim.
Worked example
All 5 observations equal 4. Population SD? all deviations = 0 = 0
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
120. Comparing SD conventions
Level 4 · Project: a defensible report
Report method, count, units and limitations. Do not choose summaries merely to support a claim.
Worked example
n=4, population variance 3. Sample variance of the same data? 3 × 4/3 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
Test: choose your scope
Choose a lesson range and test length. Topics are drawn without repetition, up to the number available in that range. Answers appear after the test; there is no time limit.
Further reading
Original exercises by Pewniaki. The mental-maths identities are explained algebraically; no claim of historical origin or guaranteed speed advantage is made.