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14.5."},{"text":"10 and 14","correct":false,"explanation":"Using 10 and 14 gives the stated class limits rather than the true boundaries. Rounding to whole numbers extends each outer limit by 0.5, giving 9.5 and 14.5."},{"text":"9.5 and 13.5","correct":false,"explanation":"Both boundaries have been shifted down by 0.5. Only the lower limit decreases; the upper limit increases, so the true boundaries are 9.5 and 14.5."},{"text":"10.5 and 14.5","correct":false,"explanation":"Both boundaries have been shifted up by 0.5. The lower boundary must instead be reduced by 0.5, so the true boundaries are 9.5 and 14.5."}]},{"id":"53025306-929e-4dbe-970c-9b4be8e85010","cardType":"basic","points":1,"front":"In a histogram, what does the area of each bar represent?","back":"Frequency","mcqOptions":[{"text":"Frequency","correct":true,"explanation":"A histogram is constructed so that each bar’s area equals, or is proportional to, the class frequency. Its height is therefore the frequency density."},{"text":"Frequency density","correct":false,"explanation":"Frequency density is represented by bar height, not area. Multiplying frequency density by class width gives the bar’s area, which represents frequency."},{"text":"Class width","correct":false,"explanation":"Class width is represented by the horizontal width of a bar, not its area. The bar’s area represents the frequency in that class."},{"text":"Cumulative frequency","correct":false,"explanation":"Cumulative frequency is represented on a cumulative frequency graph, not by individual histogram bars. In a histogram, each bar’s area represents its class frequency."}]},{"id":"72515c53-913c-4a41-8bc4-bba3f756f55a","cardType":"basic","points":1,"front":"What is the formula for frequency density?","back":"$$$\\frac{\\text{frequency}}{\\text{class width}}$$","mcqOptions":[{"text":"$$$\\frac{\\text{frequency}}{\\text{class width}}$$","correct":true,"explanation":"Frequency density is found by dividing the class frequency by its class width. This ensures that bar area, calculated as width times density, represents frequency."},{"text":"$$$\\frac{\\text{class width}}{\\text{frequency}}$$","correct":false,"explanation":"This reverses the required division. Frequency density is frequency divided by class width, ensuring that multiplying density by class width recovers the frequency."},{"text":"$$$\\text{frequency} \\times \\text{class width}$$","correct":false,"explanation":"Multiplying frequency by class width does not give a density and makes wider classes disproportionately tall. The correct formula divides frequency by class width."},{"text":"$$$\\frac{\\text{frequency}}{\\text{class midpoint}}$$","correct":false,"explanation":"Dividing frequency by class midpoint confuses a class’s location with its width. Frequency density is calculated using frequency divided by the class width."}]},{"id":"f0450ef1-b310-44a4-9131-93c56b6f1d81","cardType":"cloze","points":null,"front":"In a histogram, the vertical axis represents [ ], not frequency.","back":"In a histogram, the vertical axis represents frequency density, not frequency.","mcqOptions":[{"text":"frequency density","correct":true,"explanation":"Histogram bar area represents frequency, so the height must be frequency divided by class width. Therefore, the vertical axis displays frequency density."},{"text":"frequency","correct":false,"explanation":"Frequency is represented by the area of each histogram bar, not directly by its height. The vertical axis shows frequency density to account for unequal class widths."},{"text":"cumulative frequency","correct":false,"explanation":"Cumulative frequency belongs on the vertical axis of a cumulative frequency graph. A histogram’s vertical axis instead represents frequency density."},{"text":"class width","correct":false,"explanation":"Class width is measured along the horizontal axis because it describes each interval’s extent. The histogram’s vertical axis represents frequency density."}]},{"id":"b3e5b904-58cc-49fe-896f-90d8b8c2b360","cardType":"basic","points":2,"front":"How do you calculate the horizontal scale on a drawn histogram (cm per unit)?","back":"$$$\\frac{\\text{drawn width in cm}}{\\text{true class width}}$$","mcqOptions":[{"text":"$$$\\frac{\\text{drawn width in cm}}{\\text{true class width}}$$","correct":true,"explanation":"The scale in centimetres per unit compares the physical width on the drawing with the corresponding true class width, so drawn width is divided by true width."},{"text":"$$$\\frac{\\text{true class width}}{\\text{drawn width in cm}}$$","correct":false,"explanation":"This is the reciprocal and gives units per centimetre rather than centimetres per unit. For cm per unit, divide drawn width by true class width."},{"text":"$$$\\frac{\\text{drawn width in cm}}{\\text{class frequency}}$$","correct":false,"explanation":"Frequency determines bar area but does not determine the horizontal scale. The horizontal scale is drawn width in centimetres divided by the true class width."},{"text":"$$$\\frac{\\text{drawn width in cm}}{\\text{frequency density}}$$","correct":false,"explanation":"Frequency density determines bar height rather than horizontal width. The horizontal scale must compare drawn width in centimetres with the corresponding true class width."}]},{"id":"5d4adf5a-c27f-4d08-b87f-5d68b155e510","cardType":"basic","points":2,"front":"Why are histogram height and width scales calculated independently?","back":"Frequency is represented by area, so height must represent frequency density specifically.","mcqOptions":[{"text":"Frequency is represented by area, so height must represent frequency density specifically.","correct":true,"explanation":"Histogram frequency is encoded by bar area. Once the horizontal scale represents class width, the independent vertical scale must represent frequency density so areas remain proportional to frequencies."},{"text":"Frequency is represented by height, so width must represent class frequency specifically.","correct":false,"explanation":"Frequency is not represented by height alone unless every class has equal width. Histogram area represents frequency, so height must specifically represent frequency density."},{"text":"Frequency is represented by width, so height must represent cumulative frequency specifically.","correct":false,"explanation":"Class width is shown horizontally, but area rather than width represents frequency. The independent height scale must therefore measure frequency density, not cumulative frequency."},{"text":"Frequency density is represented by area, so height must represent frequency specifically.","correct":false,"explanation":"Frequency density is represented by bar height, not area. Bar area represents frequency because multiplying frequency density by class width recovers the class frequency."}]},{"id":"89addf6f-50a7-4d9a-ac4e-7073edd28504","cardType":"cloze","points":null,"front":"The interquartile range (IQR) represents the spread of the middle [ ] of the data.","back":"The interquartile range (IQR) represents the spread of the middle 50% of the data.","mcqOptions":[{"text":"50%","correct":true,"explanation":"The IQR runs from the lower quartile at the 25th percentile to the upper quartile at the 75th percentile, containing the middle 50% of observations."},{"text":"25%","correct":false,"explanation":"A single quartile contains 25% of the data, but the interval from the lower to the upper quartile spans two quartiles and therefore contains 50%."},{"text":"75%","correct":false,"explanation":"The upper quartile marks the point below which 75% of observations lie, but the IQR starts at the lower quartile and contains only the middle 50%."},{"text":"100%","correct":false,"explanation":"The range describes the spread of all 100% of the data. The IQR excludes the lowest and highest quarters, leaving the middle 50%."}]},{"id":"29cc3768-4ede-42bb-ac1d-8a9b420514c7","cardType":"basic","points":1,"front":"How do you calculate the IQR from quartiles?","back":"$$Q_3 - Q_1$","mcqOptions":[{"text":"$$Q_3 - Q_1$","correct":true,"explanation":"The interquartile range is the distance from the lower quartile to the upper quartile, so it is calculated by subtracting $Q_1$ from $Q_3$."},{"text":"$$Q_1 - Q_3$","correct":false,"explanation":"This reverses the subtraction and normally produces a negative value. The spread must be non-negative, so calculate the upper quartile minus the lower quartile: $Q_3 - Q_1$."},{"text":"$$Q_3 + Q_1$","correct":false,"explanation":"Adding the quartiles does not measure the distance between them. The IQR is the difference between the upper and lower quartiles, $Q_3 - Q_1$."},{"text":"$$Q_3 - Q_2$","correct":false,"explanation":"Subtracting the median from the upper quartile measures only the upper half of the central interval. The full IQR is $Q_3 - Q_1$."}]},{"id":"3b7a1fc2-d5b0-4089-b644-788979a9d1f3","cardType":"basic","points":1,"front":"What is the lower fence formula for identifying outliers?","back":"$$Q_1 - 1.5 \\times \\text{IQR}$","mcqOptions":[{"text":"$$Q_1 - 1.5 \\times \\text{IQR}$","correct":true,"explanation":"The lower fence is found by subtracting 1.5 times the interquartile range from the first quartile. Values below this boundary are potential low outliers."},{"text":"$$Q_1 + 1.5 \\times \\text{IQR}$","correct":false,"explanation":"Adding 1.5 times the IQR gives the upper fence, not the lower fence. The lower fence is $Q_1 - 1.5 \\times \\text{IQR}$, identifying unusually small values."},{"text":"$$Q_3 - 1.5 \\times \\text{IQR}$","correct":false,"explanation":"Using $Q_3$ starts from the third quartile, so this does not define the lower boundary. The lower fence uses $Q_1$ and subtracts 1.5 times the IQR."},{"text":"$$Q_1 - 1.5 \\times \\text{range}$","correct":false,"explanation":"The outlier rule uses the interquartile range, which measures the middle 50% of the data, rather than the full range. Therefore the correct formula is $Q_1 - 1.5 \\times \\text{IQR}$."}]},{"id":"bcbe34bf-5fd9-4352-b23d-10e7b4a398a5","cardType":"basic","points":1,"front":"What is the upper fence formula for identifying outliers?","back":"$$Q_3 + 1.5 \\times \\text{IQR}$","mcqOptions":[{"text":"$$Q_3 + 1.5 \\times \\text{IQR}$","correct":true,"explanation":"The upper fence is found by adding 1.5 times the interquartile range to the third quartile. Values above this boundary are potential high outliers."},{"text":"$$Q_3 - 1.5 \\times \\text{IQR}$","correct":false,"explanation":"Subtracting 1.5 times the IQR gives a boundary below the third quartile, not the upper fence. The upper fence is $Q_3 + 1.5 \\times \\text{IQR}$."},{"text":"$$Q_1 + 1.5 \\times \\text{IQR}$","correct":false,"explanation":"The first quartile is used for the lower fence. For the upper fence, start with $Q_3$ and add 1.5 times the IQR to identify unusually large values."},{"text":"$$Q_3 + 1.5 \\times \\text{range}$","correct":false,"explanation":"The outlier rule uses the interquartile range, not the full range. The correct upper-fence calculation is $Q_3 + 1.5 \\times \\text{IQR}$."}]},{"id":"cc957a5b-b4cc-4183-91fc-fa9abd1b9518","cardType":"basic","points":1,"front":"Where do the whiskers stop on a box plot if outliers are present?","back":"At the smallest and largest non-outlier values.","mcqOptions":[{"text":"At the smallest and largest non-outlier values.","correct":true,"explanation":"Whiskers extend to the most extreme observations that are not outliers. Separate symbols show values beyond the lower or upper fences."},{"text":"At the smallest and largest outlier values.","correct":false,"explanation":"Outliers are plotted separately and are not included in the whisker endpoints. Each whisker stops at the smallest or largest non-outlier value."},{"text":"At the lower and upper quartiles.","correct":false,"explanation":"The quartiles form the ends of the box, not the whiskers. When outliers exist, the whiskers extend from the box to the most extreme non-outlier values."},{"text":"At the minimum and maximum data values.","correct":false,"explanation":"The overall minimum or maximum may be an outlier, so it is plotted separately rather than reached by a whisker. Whiskers stop at the extreme non-outliers."}]},{"id":"ee398a28-a7a1-4f64-a310-a85ea335b972","cardType":"cloze","points":null,"front":"On a box plot, outliers are typically plotted separately using [ ].","back":"On a box plot, outliers are typically plotted separately using crosses.","mcqOptions":[{"text":"crosses","correct":true,"explanation":"Crosses are commonly used as separate symbols for observations beyond the box plot fences, allowing outliers to be distinguished from the whiskers."},{"text":"circles","correct":false,"explanation":"Although circles can mark points in other graphs, the convention specified here uses crosses for box-plot outliers. The whiskers already identify the non-outlier extremes."},{"text":"triangles","correct":false,"explanation":"Triangles are not the standard symbol expected for separately plotted box-plot outliers in this convention. Crosses identify observations beyond the fences."},{"text":"squares","correct":false,"explanation":"Squares may be used for other plotted markers, but they are not the usual symbol for box-plot outliers here. Crosses are used to show those separate values."}]},{"id":"37313a78-4e1c-469b-9447-70c4906f5c5d","cardType":"basic","points":3,"front":"List three features to compare when describing two distributions using box plots.","back":"

Location (compare medians)
Spread (compare IQRs or ranges)
Outliers (mention any in context)

","mcqOptions":[{"text":"

Location (compare medians)
Spread (compare IQRs or ranges)
Outliers (mention any in context)

","correct":true,"explanation":"A box-plot comparison should discuss typical value using medians, variation using IQRs or ranges, and any outliers in the context of the data."},{"text":"

Location (compare means)
Spread (compare standard deviations)
Frequency (compare totals)

","correct":false,"explanation":"Box plots directly show medians, IQRs or ranges, and outliers; they do not show means, standard deviations, or frequency totals. Those are not the appropriate three features here."},{"text":"

Centre (compare modes)
Shape (compare histograms)
Frequency (compare totals)

","correct":false,"explanation":"Modes, histogram shape, and frequency totals cannot be read reliably from a box plot. The relevant comparison uses medians, spread measures, and outliers."},{"text":"

Position (compare minima)
Variation (compare ranges only)
Frequency (compare class widths)

","correct":false,"explanation":"Minimum values alone do not describe location, and box plots do not display frequencies or class widths. Compare medians, IQRs or ranges, and outliers instead."}]},{"id":"6f5cf900-37b7-432d-9712-95e4a52a745b","cardType":"basic","points":2,"front":"When are the median and IQR preferred over the mean and standard deviation?","back":"When data are skewed or contain outliers.","mcqOptions":[{"text":"When data are skewed or contain outliers.","correct":true,"explanation":"The median and IQR are resistant to extreme values and are therefore more representative than the mean and standard deviation when data are skewed or contain outliers."},{"text":"When data are symmetric and contain no outliers.","correct":false,"explanation":"For symmetric data without outliers, the mean and standard deviation are usually suitable. The median and IQR are preferred when skewness or outliers would distort those measures."},{"text":"When data are categorical or contain no numerical values.","correct":false,"explanation":"The median and IQR require ordered numerical data, so they are not suitable for purely categorical data. They are preferred for numerical data affected by skewness or outliers."},{"text":"When data have a large sample size and contain no outliers.","correct":false,"explanation":"A large sample size does not by itself make the median and IQR preferable. With no outliers, the mean and standard deviation are often appropriate, while skewness or outliers favour the median and IQR."}]},{"id":"71e429c8-c259-47c0-b195-868e93f9fe38","cardType":"basic","points":1,"front":"What is the formula for standard deviation using summary statistics?","back":"$$$\\sqrt{\\frac{\\sum x^2}{n} - \\bar{x}^2}$$","mcqOptions":[{"text":"$$$\\sqrt{\\frac{\\sum x^2}{n} - \\bar{x}^2}$$","correct":true,"explanation":"The standard deviation is the square root of the mean of the squared values minus the square of the mean: $\\sqrt{\\frac{\\sum x^2}{n} - \\bar{x}^2}$."},{"text":"$$$\\sqrt{\\frac{\\sum x^2}{n} + \\bar{x}^2}$$","correct":false,"explanation":"The square of the mean must be subtracted, not added. Adding it makes the variance too large and does not produce the standard deviation formula."},{"text":"$$$\\sqrt{\\frac{\\sum x^2}{n-1} - \\bar{x}^2}$$","correct":false,"explanation":"Dividing by $n-1$ is associated with an unbiased sample estimate in some statistical contexts. The stated summary-statistics formula for standard deviation divides by $n$."},{"text":"$$$\\sqrt{\\frac{\\sum x^2}{n} - \\bar{x}}$$","correct":false,"explanation":"The mean must be squared before it is subtracted from the mean of the squared values. Omitting that square gives an incorrect variance and hence an incorrect standard deviation."}]},{"id":"17d4f311-909d-4a00-9255-52c4e21c4592","cardType":"cloze","points":null,"front":"A variable is measured to the nearest whole number. The class 11-18 has true boundaries of [ ] and 18.5.","back":"A variable is measured to the nearest whole number. The class 11-18 has true boundaries of 10.5 and 18.5.","mcqOptions":[{"text":"10.5","correct":true,"explanation":"Measurements rounded to the nearest whole number vary by 0.5 on either side of the stated value. Therefore the lower boundary for 11 is $11 - 0.5 = 10.5$."},{"text":"10","correct":false,"explanation":"Using 10 treats the stated class limit as an exact boundary rather than a rounded measurement. Rounding to the nearest whole number requires subtracting 0.5, giving 10.5."},{"text":"10.4","correct":false,"explanation":"A half-unit adjustment is required for values measured to the nearest whole number, not a 0.6-unit adjustment. The lower true boundary is $11 - 0.5 = 10.5$."},{"text":"11.5","correct":false,"explanation":"Adding 0.5 to 11 gives the upper boundary, not the lower one. The lower boundary is found by subtracting 0.5, so it is 10.5."}]},{"id":"c418c9c5-0ecc-4b8c-9c73-75027a10848b","cardType":"cloze","points":null,"front":"A variable is measured to the nearest whole number. The class 11-18 has true boundaries of 10.5 and [ ].","back":"A variable is measured to the nearest whole number. The class 11-18 has true boundaries of 10.5 and 18.5.","mcqOptions":[{"text":"18.5","correct":true,"explanation":"Values recorded as 18 represent measurements from 17.5 up to, but not including, 18.5. Therefore, the upper true class boundary is 18.5."},{"text":"18","correct":false,"explanation":"Using 18 treats the stated class limit as the true boundary. Measurements are rounded to the nearest whole number, so the upper boundary is half a unit higher at 18.5."},{"text":"19","correct":false,"explanation":"Adding a whole unit instead of half a unit gives 19. For measurements rounded to the nearest integer, the upper true boundary is $18 + 0.5 = 18.5$."},{"text":"17.5","correct":false,"explanation":"Subtracting 0.5 from the upper class limit gives 17.5, which is the lower boundary for values recorded as 18. The class’s upper boundary is 18.5."}]},{"id":"70844e78-66af-4b89-99c6-d0f2b5bc48ef","cardType":"cloze","points":null,"front":"When finding quartiles from an odd number of data values, do not include the [ ] in either half for calculation.","back":"When finding quartiles from an odd number of data values, do not include the median in either half for calculation.","mcqOptions":[{"text":"median","correct":true,"explanation":"With an odd number of ordered values, the median is the single central value. Excluding it leaves two equal halves from which the lower and upper quartiles are found."},{"text":"lower quartile","correct":false,"explanation":"The lower quartile is calculated from the lower half rather than removed before splitting the data. The central median must be excluded from both halves."},{"text":"upper quartile","correct":false,"explanation":"The upper quartile is calculated from the upper half, so it is not the value omitted when dividing the data. The central median is excluded from both halves."},{"text":"mean","correct":false,"explanation":"The mean need not occupy the central position in ordered data and is not removed during quartile calculation. For an odd-sized data set, the median is excluded."}]},{"id":"87f669f2-9ab4-4292-97aa-9635065263a7","cardType":"basic","points":2,"front":"How do you estimate probability from a histogram if the total frequency is known?","back":"$$$\\frac{\\text{area of required section}}{\\text{total area}} \\times \\text{total frequency}$$ (then divide by total frequency) or simply the ratio of areas.","mcqOptions":[{"text":"$$$\\frac{\\text{area of required section}}{\\text{total area}} \\times \\text{total frequency}$$ (then divide by total frequency) or simply the ratio of areas.","correct":true,"explanation":"Histogram frequency is proportional to bar area because area equals class width multiplied by frequency density. Therefore, the required area divided by total area gives the estimated probability."},{"text":"$$$\\frac{\\text{height of required section}}{\\text{total height}} \\times \\text{total frequency}$$ (then divide by total frequency) or simply the ratio of heights.","correct":false,"explanation":"Bar height represents frequency density, not frequency, so height ratios fail when class widths differ. Estimated probability must be found using the required area divided by the total area."},{"text":"$$$\\frac{\\text{width of required section}}{\\text{total width}} \\times \\text{total frequency}$$ (then divide by total frequency) or simply the ratio of widths.","correct":false,"explanation":"Class width alone ignores frequency density, so width ratios do not generally represent frequencies. Histogram probability is estimated from the required section’s area as a proportion of total area."},{"text":"$$$1-\\frac{\\text{area of required section}}{\\text{total area}} \\times \\text{total frequency}$$ (then divide by total frequency) or simply one minus the ratio of areas.","correct":false,"explanation":"Subtracting the area ratio from 1 gives the probability of being outside the required section, not inside it. The desired probability is simply required area divided by total area."}]},{"id":"2f00563a-cbb5-469d-8b55-4dead8a5da01","cardType":"cloze","points":null,"front":"Mean and standard deviation are most useful when data are [ ] and have no major outliers.","back":"Mean and standard deviation are most useful when data are symmetrical and have no major outliers.","mcqOptions":[{"text":"symmetrical","correct":true,"explanation":"For symmetrical data without major outliers, the mean represents the centre well and the standard deviation describes spread around it without being unduly distorted by extreme values."},{"text":"skewed","correct":false,"explanation":"In skewed data, the long tail pulls the mean and can inflate the standard deviation. The median and interquartile range are usually preferable; mean and standard deviation suit symmetrical data."},{"text":"qualitative","correct":false,"explanation":"Qualitative data consist of categories rather than meaningful numerical measurements, so calculating a mean and standard deviation is generally invalid. These measures are useful for symmetrical quantitative data."},{"text":"ranked","correct":false,"explanation":"Ranked data show order but do not guarantee equal numerical intervals, making the mean and standard deviation inappropriate. They are most useful for symmetrical quantitative data without major outliers."}]}]}],"examCountdown":{"examDate":"2027-05-13","paperLabel":"Paper 1"},"heading":"Representations of Data","title":"Representations of Data Flashcards","topbar":{"backHref":"/as-level/maths/edexcel/flashcards"},"topicSidebar":"$L37"}]}],"topicId":"b2c8a667-b21a-41fc-a29f-94a525921f3f","walled":"$L38"}],"$L39","$L3a"]