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22:[["$","$L31",null,{"feature":"flashcards"}],"$L32",["$","$L33",null,{"href":"/as-level/maths/edexcel/as-statistics-histograms/flashcards","label":"3.4 Histograms","subjectCode":"as_level_maths","surface":"flashcards"}],["$","$L34",null,{"runner":["$","div",null,{"className":"min-h-0 flex-1 overflow-hidden bg-background","children":["$","$L35",null,{"decks":[{"id":"cac7f5c8-2081-4073-a66e-518807495572","title":"Histograms Flashcards","topicId":"cb90542c-a7eb-46c8-8be7-31dc51cd2479","topicSlug":"as-statistics-histograms","cards":[{"id":"04febb36-5451-418a-b16e-97b521cdc8f3","cardType":"basic","points":1,"front":"In a histogram, what does the area of each bar represent?","back":"The frequency of the class.","mcqOptions":[{"text":"The frequency of the class.","correct":true,"explanation":"A histogram is scaled so that each bar’s area, given by class width multiplied by frequency density, equals the frequency in that class."},{"text":"The frequency density of the class.","correct":false,"explanation":"Frequency density determines the bar’s height, not its area. The area represents the class frequency because width multiplied by frequency density equals frequency."},{"text":"The class width of the class.","correct":false,"explanation":"Class width is represented by the horizontal span of the bar, not its area. The bar’s area represents the frequency of the class."},{"text":"The cumulative frequency of the class.","correct":false,"explanation":"Cumulative frequency is the running total across classes and is usually shown on a cumulative frequency graph. In a histogram, each bar’s area represents its class frequency."}]},{"id":"24b34d60-46ab-495e-b7ec-e60a259fd63f","cardType":"cloze","points":null,"front":"The formula for frequency density is $$\\text{[...]}$$.","back":"The formula for frequency density is $$\\boxed{\\text{frequency density} = \\frac{\\text{frequency}}{\\text{class width}}}$$.","mcqOptions":[{"text":"$$$\\text{frequency density} = \\frac{\\text{frequency}}{\\text{class width}}$$","correct":true,"explanation":"Dividing frequency by class width gives frequency per unit of the horizontal scale, ensuring that the bar’s area equals the class frequency."},{"text":"$$$\\text{frequency density} = \\text{frequency} \\times \\text{class width}$$","correct":false,"explanation":"Multiplying by class width calculates an area-like quantity rather than frequency density. Frequency density is found by dividing frequency by class width."},{"text":"$$$\\text{frequency density} = \\frac{\\text{class width}}{\\text{frequency}}$$","correct":false,"explanation":"Reversing the fraction gives class width per observation, not frequency per unit width. The correct numerator is frequency and the denominator is class width."},{"text":"$$$\\text{frequency density} = \\frac{\\text{frequency}}{\\text{upper class boundary}}$$","correct":false,"explanation":"Dividing by the upper class boundary ignores the interval’s lower boundary and therefore its width. Frequency density is frequency divided by the full class width."}]},{"id":"53dc99a6-a68e-483a-849b-f89f15eff60b","cardType":"basic","points":1,"front":"What is represented on the vertical axis (y-axis) of a histogram?","back":"Frequency density.","mcqOptions":[{"text":"Frequency density.","correct":true,"explanation":"A histogram’s vertical scale shows frequency per unit class width. This frequency density makes each bar’s area equal to the corresponding class frequency."},{"text":"Frequency.","correct":false,"explanation":"Frequency is represented by the area of a histogram bar, not generally its height. The vertical axis shows frequency density, particularly when class widths differ."},{"text":"Class width.","correct":false,"explanation":"Class width is displayed along the horizontal axis through the span of each interval. The vertical axis of a histogram represents frequency density."},{"text":"Cumulative frequency.","correct":false,"explanation":"Cumulative frequency is used as the vertical variable on a cumulative frequency graph, not a histogram. A histogram’s vertical axis represents frequency density."}]},{"id":"4dfbfa29-f427-47d8-a9d9-5a988e4b222f","cardType":"cloze","points":null,"front":"If data is measured to the nearest whole number, the class width of the interval 10–14 is [ ].","back":"If data is measured to the nearest whole number, the class width of the interval 10–14 is 5.","mcqOptions":[{"text":"5","correct":true,"explanation":"Values recorded from 10 to 14 have continuous class boundaries 9.5 and 14.5. Their difference is $14.5-9.5=5$, so the class width is 5."},{"text":"4","correct":false,"explanation":"Subtracting the stated endpoints gives $14-10=4$, but this ignores the half-unit class boundaries. Using 9.5 and 14.5 gives the correct width of 5."},{"text":"6","correct":false,"explanation":"Counting one extra unit beyond both endpoints overstates the interval. The correct boundaries are 9.5 and 14.5, whose difference gives a class width of 5."},{"text":"4.5","correct":false,"explanation":"Adding only one half-unit to the endpoint difference accounts for just one boundary. Both boundaries must be adjusted, giving $14.5-9.5=5$."}]},{"id":"5d55b858-ebe9-4eda-9da2-bc9455f4115e","cardType":"basic","points":1,"front":"Why do the bars in a histogram typically touch?","back":"The data is continuous (or treated as continuous).","mcqOptions":[{"text":"The data is continuous (or treated as continuous).","correct":true,"explanation":"Histograms group continuous data into adjacent class intervals, so there are no categorical gaps between neighbouring classes and the bars touch."},{"text":"The data is discrete (or treated as discrete).","correct":false,"explanation":"Discrete data consists of separate values and is often displayed with gaps in a bar chart. Histogram bars touch because the data is continuous or treated as continuous."},{"text":"The data is categorical (or treated as categorical).","correct":false,"explanation":"Categorical data is normally represented by a bar chart whose separated bars distinguish categories. Histogram bars touch because their intervals contain continuous data."},{"text":"The data is cumulative (or treated as cumulative).","correct":false,"explanation":"Cumulative data concerns running totals and does not explain adjacent bars. Histogram bars touch because the underlying variable is continuous or is being treated as continuous."}]},{"id":"146f89ed-482c-4f72-990e-989b680d5782","cardType":"basic","points":2,"front":"How is the class width calculated for the interval 9–13 when data is rounded to the nearest integer?","back":"

Use class boundaries 8.5 to 13.5
$13.5 - 8.5 = 5$

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

Use class boundaries 8.5 to 13.5
$13.5 - 8.5 = 5$

","correct":true,"explanation":"Rounding to the nearest integer means the recorded endpoints correspond to boundaries half a unit below and above. Thus the width is $13.5-8.5=5$."},{"text":"

Use class limits 9 to 13
$13 - 9 = 4$

","correct":false,"explanation":"Subtracting the recorded endpoints ignores the half-unit regions included through rounding. The true boundaries are 8.5 and 13.5, so the class width is 5."},{"text":"

Use class boundaries 9.5 to 13.5
$13.5 - 9.5 = 4$

","correct":false,"explanation":"The lower boundary must be 8.5, not 9.5, because values from 8.5 round to 9. Using 8.5 and 13.5 gives width 5."},{"text":"

Use class boundaries 8 to 14
$14 - 8 = 6$

","correct":false,"explanation":"Extending each endpoint by a whole unit double-counts the rounding allowance. Nearest-integer boundaries lie half a unit away, giving 8.5 to 13.5 and width 5."}]},{"id":"74fd0b93-f931-40db-853a-71b8d4fa2bac","cardType":"basic","points":2,"front":"If two classes have the same frequency but class A is wider than class B, which bar is taller?","back":"Class B (the narrower class has a higher frequency density).","mcqOptions":[{"text":"Class B (the narrower class has a higher frequency density).","correct":true,"explanation":"Frequency density equals frequency divided by class width. With equal frequencies, the narrower class B has the greater density and therefore the taller bar."},{"text":"Class A (the wider class has a higher frequency density).","correct":false,"explanation":"A greater class width lowers frequency density when frequency is fixed, so width does not make class A taller. The narrower class B has the taller bar."},{"text":"Neither class (the two classes have the same frequency density).","correct":false,"explanation":"Equal frequencies give equal bar areas, not equal heights. Because class B is narrower, it requires a greater frequency density and therefore a taller bar."},{"text":"It cannot be determined (the frequencies alone do not determine bar height).","correct":false,"explanation":"The given frequencies and widths are sufficient because height represents frequency density. Dividing the same frequency by a smaller width makes class B’s bar taller."}]},{"id":"0b939afa-e433-49b8-a985-2f02b14bd5c1","cardType":"basic","points":1,"front":"How do you calculate frequency from a histogram bar?","back":"$$\\text{class width} \\times \\text{frequency density}$","mcqOptions":[{"text":"$$\\text{class width} \\times \\text{frequency density}$","correct":true,"explanation":"A histogram bar’s area represents frequency. Since area is width multiplied by height, frequency equals class width multiplied by frequency density."},{"text":"$$\\text{class width} \\div \\text{frequency density}$","correct":false,"explanation":"Dividing width by density does not calculate a bar’s area. Frequency is found by multiplying the class width by the frequency density."},{"text":"$$\\text{frequency density} \\div \\text{class width}$","correct":false,"explanation":"Dividing density by width applies the width adjustment a second time rather than recovering frequency. Multiply frequency density by class width instead."},{"text":"$$\\text{class width} + \\text{frequency density}$","correct":false,"explanation":"Adding the bar’s width and height does not calculate its area. Frequency is represented by area, so class width and frequency density must be multiplied."}]},{"id":"b050f2f0-0fcb-4d55-8228-c03a8833817d","cardType":"cloze","points":null,"front":"When a histogram bar is measured in cm on paper, the horizontal scale links class width to [ ].","back":"When a histogram bar is measured in cm on paper, the horizontal scale links class width to drawn width.","mcqOptions":[{"text":"drawn width","correct":true,"explanation":"The horizontal scale converts a class width measured in data units into the physical width used to draw that class on the histogram."},{"text":"drawn height","correct":false,"explanation":"Drawn height is determined by frequency density and the vertical scale. The horizontal scale instead links each class width to its drawn width."},{"text":"bar area","correct":false,"explanation":"Bar area represents frequency after both scales are considered, so it is not set by the horizontal scale alone. Class width is linked to drawn width."},{"text":"frequency density","correct":false,"explanation":"Frequency density determines a bar’s height rather than its horizontal dimension. The horizontal scale converts class width into drawn width on the paper."}]},{"id":"378780fa-685c-4d5a-b4f6-2cc408d94759","cardType":"cloze","points":null,"front":"When a histogram bar is measured in cm on paper, the vertical scale links frequency density to [ ].","back":"When a histogram bar is measured in cm on paper, the vertical scale links frequency density to drawn height.","mcqOptions":[{"text":"drawn height","correct":true,"explanation":"The vertical scale converts frequency density into the physical height of a histogram bar, just as the horizontal scale converts class width into drawn width."},{"text":"drawn width","correct":false,"explanation":"Drawn width is determined by class width and the horizontal scale. The vertical scale instead links frequency density to the bar’s drawn height."},{"text":"bar area","correct":false,"explanation":"Bar area depends on both width and height, so the vertical scale alone does not determine it. Frequency density is linked specifically to drawn height."},{"text":"class frequency","correct":false,"explanation":"Class frequency is represented by bar area, not directly by the vertical measurement. The vertical scale converts frequency density into drawn height."}]},{"id":"2880419a-d536-47ee-bc78-3799c1c74d19","cardType":"basic","points":1,"front":"A variable is measured to the nearest whole number. What are the class boundaries for the interval 20–22?","back":"19.5 and 22.5","mcqOptions":[{"text":"19.5 and 22.5","correct":true,"explanation":"Values rounded to 20 begin at 19.5, while values rounded to 22 extend up to 22.5. These are therefore the continuous class boundaries."},{"text":"20 and 22","correct":false,"explanation":"Using 20 and 22 gives the stated class limits, not its continuous boundaries. Measurements to the nearest whole number require subtracting and adding 0.5."},{"text":"20.5 and 22.5","correct":false,"explanation":"Adding 0.5 at both ends shifts the whole interval upwards. The lower boundary must be 19.5, while the upper boundary is 22.5."},{"text":"19.5 and 21.5","correct":false,"explanation":"Subtracting 0.5 from both limits makes the upper boundary too low. The interval starts at 19.5 but extends to 22.5."}]},{"id":"f22b19b4-8d28-446b-8422-39c9e8af00a0","cardType":"basic","points":2,"front":"A class 4–8 (nearest whole number) has a frequency of 10. What is its frequency density?","back":"

Class width = 5
$10 \\div 5 = 2$

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

Class width = 5
$10 \\div 5 = 2$

","correct":true,"explanation":"The continuous boundaries are 3.5 and 8.5, giving a class width of 5. Frequency density is frequency divided by class width, so it equals 2."},{"text":"

Class width = 4
$10 \\div 4 = 2.5$

","correct":false,"explanation":"A width of 4 comes from subtracting the stated integer limits and ignoring rounding boundaries. The true width is 5, so the density is 2."},{"text":"

Class width = 5
$5 \\div 10 = 0.5$

","correct":false,"explanation":"This reverses the frequency-density calculation. Frequency must be divided by class width, giving $10 \\div 5 = 2$, rather than width divided by frequency."},{"text":"

Class width = 6
$10 \\div 6 = 1.67$

","correct":false,"explanation":"A width of 6 incorrectly extends the continuous boundaries by a whole unit at each end. The boundaries are 3.5 and 8.5, giving width 5 and density 2."}]},{"id":"f3ad4d0e-c7ef-42e7-a742-36aa8ef0197f","cardType":"cloze","points":null,"front":"To estimate the probability of an event from a histogram, you divide the [ ] by the total frequency.","back":"To estimate the probability of an event from a histogram, you divide the estimated frequency for the event by the total frequency.","mcqOptions":[{"text":"estimated frequency for the event","correct":true,"explanation":"Estimated probability is relative frequency: estimate how many observations satisfy the event, then divide that estimated frequency by the frequency of the entire data set."},{"text":"frequency density for the event","correct":false,"explanation":"Frequency density is a frequency per unit class width, not the number of observations satisfying an event. The numerator must be the estimated frequency for the event."},{"text":"estimated area for the event","correct":false,"explanation":"Histogram area is proportional to frequency but may not equal it numerically because of the plotting scales. The numerator paired with total frequency is estimated event frequency."},{"text":"class width covered by the event","correct":false,"explanation":"Class width measures an interval on the horizontal axis rather than how many observations meet the event. The required numerator is the estimated frequency for the event."}]},{"id":"d78800d3-a522-4999-98ac-a7602cdaf3ac","cardType":"cloze","points":null,"front":"To estimate the probability of an event from a histogram, you divide the estimated frequency for the event by the [ ].","back":"To estimate the probability of an event from a histogram, you divide the estimated frequency for the event by the total frequency.","mcqOptions":[{"text":"total frequency","correct":true,"explanation":"A probability is the relevant frequency divided by the frequency of all possible outcomes. Here, the event’s estimated frequency is therefore divided by total frequency."},{"text":"total class width","correct":false,"explanation":"Total class width measures the combined horizontal range and does not count observations. Estimated event frequency must be divided by total frequency to obtain a probability."},{"text":"total frequency density","correct":false,"explanation":"Frequency densities cannot be added as though they were frequencies when class widths differ. The denominator of the relative-frequency calculation is the total frequency."},{"text":"total area of the bars","correct":false,"explanation":"Total bar area is proportional to total frequency but is not generally numerically equal to it because graph scales may vary. The required denominator is total frequency."}]},{"id":"8db59a2a-d40c-479a-83eb-cb6bcee11491","cardType":"basic","points":1,"front":"What assumption is made when a probability boundary cuts through a histogram class?","back":"Data is evenly spread (distributed) within that class.","mcqOptions":[{"text":"Data is evenly spread (distributed) within that class.","correct":true,"explanation":"A histogram gives no detail about positions within a class. Estimating a partial class therefore assumes observations are uniformly distributed across that class interval."},{"text":"Data is concentrated at the midpoint within that class.","correct":false,"explanation":"Assuming all observations lie at the midpoint would make a partial-class estimate all-or-nothing. Instead, data is assumed to be evenly distributed throughout the class."},{"text":"Data is concentrated at the boundaries within that class.","correct":false,"explanation":"Class boundaries describe the interval but do not imply observations cluster at its ends. Partial-class calculations assume data is evenly spread within the class."},{"text":"Data becomes more frequent towards the upper end of that class.","correct":false,"explanation":"A histogram does not show an increasing pattern within a single class because its bar has constant height. The estimate assumes an even distribution within the class."}]},{"id":"a3bba68e-3f51-4e89-ac7a-2273ceb8d81b","cardType":"basic","points":2,"front":"How do you estimate the frequency of a partial class where the boundary cuts through it?","back":"Take the fraction of the width covered by the event and multiply by the total class frequency.","mcqOptions":[{"text":"Take the fraction of the width covered by the event and multiply by the total class frequency.","correct":true,"explanation":"Assuming observations are uniformly distributed within the class, the proportion of its horizontal width included equals the proportion of its frequency included."},{"text":"Take the fraction of the height covered by the event and multiply by the total class frequency.","correct":false,"explanation":"The cut is made along the horizontal class interval, so the relevant proportion comes from width, not height. Multiply that width fraction by total class frequency."},{"text":"Take the fraction of the width covered by the event and multiply by the frequency density.","correct":false,"explanation":"Frequency density is a rate per unit width, not the class total. The width fraction must be multiplied by total class frequency to estimate the partial frequency."},{"text":"Take the fraction of the width not covered by the event and multiply by the total class frequency.","correct":false,"explanation":"Using the uncovered width estimates the frequency outside the event within that class. The event frequency uses the fraction of width that is covered."}]},{"id":"f3aa70b4-8aab-40fc-99ce-1b5deb169c3d","cardType":"basic","points":1,"front":"A bar has a width of 5 units and a frequency density of 2.8. What is the frequency?","back":"14 ($5 \\times 2.8$)","mcqOptions":[{"text":"14 ($5 \\times 2.8$)","correct":true,"explanation":"In a histogram, frequency equals class width multiplied by frequency density. Therefore, $5 \\times 2.8 = 14$, so the bar represents 14 observations."},{"text":"7.8 ($5 + 2.8$)","correct":false,"explanation":"Adding the class width and frequency density confuses area with a simple total. Frequency is the bar’s area, so $5 \\times 2.8 = 14$."},{"text":"0.56 ($2.8 \\div 5$)","correct":false,"explanation":"Dividing density by width reverses the histogram area calculation. Frequency is class width multiplied by frequency density, giving $5 \\times 2.8 = 14$."},{"text":"15 ($5 \\times 3$)","correct":false,"explanation":"Rounding the frequency density to 3 before calculating loses necessary accuracy. Using the given density gives $5 \\times 2.8 = 14$, not 15."}]},{"id":"635475d5-ccbb-4592-b74e-d1be87aec932","cardType":"cloze","points":null,"front":"The probability calculated from a histogram where boundaries cut through classes is only an [ ] because the exact distribution of values within classes is unknown.","back":"The probability calculated from a histogram where boundaries cut through classes is only an estimate because the exact distribution of values within classes is unknown.","mcqOptions":[{"text":"estimate","correct":true,"explanation":"A histogram does not show how values are distributed inside each class. Assuming a uniform distribution permits only an estimate when a boundary divides a class."},{"text":"exact probability","correct":false,"explanation":"The result cannot be an exact probability because the positions of individual values within each class are unknown. It is only an estimate based on an assumed distribution."},{"text":"upper bound","correct":false,"explanation":"Dividing a class proportionally does not necessarily produce an upper bound; the true proportion could lie above or below it. The calculated probability is an estimate."},{"text":"observed frequency","correct":false,"explanation":"A histogram displays grouped frequencies rather than the exact observations within each class. A probability found by splitting a class is therefore an estimate, not an observed frequency."}]},{"id":"e91b27ed-25a9-4328-9c5c-03c583cdc9a7","cardType":"basic","points":2,"front":"In a histogram, the bar for 9–13 has a class width of 5 and a drawn width of 2.5 cm. What is the horizontal scale?","back":"0.5 cm per unit ($2.5 \\div 5$)","mcqOptions":[{"text":"0.5 cm per unit ($2.5 \\div 5$)","correct":true,"explanation":"The horizontal scale is drawn width divided by class width. Thus $2.5 \\div 5 = 0.5$, meaning each data unit occupies 0.5 cm."},{"text":"2 cm per unit ($5 \\div 2.5$)","correct":false,"explanation":"This reverses the division and calculates data units per centimetre rather than centimetres per unit. The requested scale is $2.5 \\div 5 = 0.5$ cm per unit."},{"text":"0.625 cm per unit ($2.5 \\div 4$)","correct":false,"explanation":"Using $13-9=4$ overlooks the stated class width of 5, which accounts for the complete grouped interval. The scale is $2.5 \\div 5 = 0.5$ cm per unit."},{"text":"12.5 cm per unit ($2.5 \\times 5$)","correct":false,"explanation":"Multiplying the two widths does not find a scale factor. Centimetres per unit requires drawn width divided by class width, so $2.5 \\div 5 = 0.5$."}]},{"id":"d2806c2b-7ca5-415b-8149-11cc6d91d06f","cardType":"basic","points":2,"front":"If frequency density 2.4 is represented by a drawn height of 3.6 cm, what is the vertical scale?","back":"1.5 cm per unit ($3.6 \\div 2.4$)","mcqOptions":[{"text":"1.5 cm per unit ($3.6 \\div 2.4$)","correct":true,"explanation":"Centimetres per frequency-density unit is found by dividing the drawn height by the represented density. Therefore, $3.6 \\div 2.4 = 1.5$ cm per unit."},{"text":"0.67 cm per unit ($2.4 \\div 3.6$)","correct":false,"explanation":"Reversing the division calculates frequency-density units per centimetre, not centimetres per unit. The requested vertical scale is $3.6 \\div 2.4 = 1.5$ cm per unit."},{"text":"6 cm per unit ($3.6 + 2.4$)","correct":false,"explanation":"Adding the drawn height and frequency density mixes two different quantities and cannot produce a scale. Dividing gives $3.6 \\div 2.4 = 1.5$ cm per unit."},{"text":"8.64 cm per unit ($3.6 \\times 2.4$)","correct":false,"explanation":"Multiplying height by density does not calculate centimetres per unit. The scale factor requires division, so the correct calculation is $3.6 \\div 2.4 = 1.5$."}]},{"id":"b3f6a3b4-70f0-4f98-8131-b9c58b934f24","cardType":"basic","points":1,"front":"What is the primary reason for using a histogram instead of a standard bar chart?","back":"The data is grouped and the class widths are unequal.","mcqOptions":[{"text":"The data is grouped and the class widths are unequal.","correct":true,"explanation":"With unequal class widths, bar height alone cannot represent frequency fairly. A histogram uses frequency density so that each bar’s area, rather than height, represents frequency."},{"text":"The data is categorical and the bars must be separated.","correct":false,"explanation":"Categorical data with separated bars is represented by a standard bar chart. Histograms are used for grouped numerical data, particularly when unequal class widths require frequency density."},{"text":"The data is ungrouped and the values are discrete.","correct":false,"explanation":"Ungrouped discrete values do not require touching histogram bars or frequency density. A histogram is preferred when numerical data is grouped into intervals with unequal class widths."},{"text":"The data is grouped and the class widths are equal.","correct":false,"explanation":"Equal class widths do not create the key need to adjust bar heights using frequency density. Histograms are especially necessary when grouped data has unequal class widths."}]}]}],"examCountdown":{"examDate":"2027-05-13","paperLabel":"Paper 1"},"heading":"3.4 Histograms","title":"Histograms Flashcards","topbar":{"backHref":"/as-level/maths/edexcel/flashcards"},"topicSidebar":"$L36"}]}],"topicId":"cb90542c-a7eb-46c8-8be7-31dc51cd2479","walled":"$L37"}],"$L38","$L39"]

3.4 Histograms Flashcards

3.4 Histograms Flashcards