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Descriptive Statistics — Assignment Prep

Mean, median, mode, range, frequency tables, and bar charts — Concept, Worked Example, and Practice for each skill.

E1 — What Is a Statistic?

A statistic is a single number that summarises a set of data. Four of the most useful ones answer four different questions: Mean — what is a "typical" value? Add all the values, then divide by how many there are. Median — what is the middle value once the list is sorted? Mode — which value appears most often? Range — how spread out is the data? Highest value minus lowest value. Key idea: mean / median / mode describe the centre; range describes the spread.

E2 — Mean, Median, Mode, and Range from a List

When you have a raw list of numbers, follow this order every time: 1 Sort the data (needed for the median; also makes the mode easier to spot). 2 Mean = sum \div count. 3 Median = the middle value. With an odd count, take the centre position. With an even count, average the two middle values. 4 Mode = the value with the highest frequency (a list can have no mode, one mode, or several modes if there is a tie). 5 Range = largest - smallest. Key idea: sort first — then mean, median, mode, and range are all straightforward.

Example 1 — Worked example — nine test scores

Scores (out of 100): 72, 85, 91, 68, 85, 77, 90, 63, 79 Step 1 — sort. 63, 68, 72, 77, 79, 85, 85, 90, 91 Step 2 — mean. Sum = 710, count = 9 : Mean = \frac{710}{9} = 78.9 (1 d.p.) Step 3 — median. Nine values \to middle position is the 5th. Median = 79 (If there had been an even count, you would average the two middle values.) Step 4 — mode. 85 appears twice; every other value appears once. Mode = 85 Step 5 — range. Range = 91 - 63 = 28 Answers: mean 78.9, median 79, mode 85, range 28 .

Practice 1 . For the data 54, 61, 58, 61, 70, 65, 49, find the mean (1 d. p .), median, mode, and range. Answers: sorted 49, 54, 58, 61, 61, 65, 70 . Mean = \frac{418}{7} = 59.7 . Median = 61 . Mode = 61 . Range = 70 - 49 = 21 .

Practice 2 . For the data 88, 76, 92, 76, 81, 85, 76, 90, find the mean, median, mode, and range. Answers: sorted 76, 76, 76, 81, 85, 88, 90, 92 . Mean = \frac{664}{8} = 83 . Median = \frac{81 + 85}{2} = 83 (even count). Mode = 76$ . R an g e = 92 - 76 = 16$.

Practice 3 — cricket runs. Runs scored off 12 balls: 0, 4, 1, 6, 2, 4, 0, 1, 4, 6, 2, 4 . Find the mean (1 d. p .), median, mode, and range. Answers: sorted 0, 0, 1, 1, 2, 2, 4, 4, 4, 4, 6, 6 . Mean = \frac{34}{12} = 2.8 . Median = \frac{2 + 4}{2} = 3 . Mode = 4 . Range = 6 - 0 = 6 .

E3 — Organising Data into a Frequency Table

A frequency table lists each unique value (or category) once and shows how many times it occurs. That works for numbers (0 pets, 1 pet, \dots) and for labels (Drive, Cut, Pull, \dots). The tally method is the same. Key idea: after you fill the table, add the frequencies — they must equal the original total count. That check catches most tally mistakes.

Example 1 — Worked example — pets owned (20 students)

Raw data: 0, 1, 2, 1, 0, 3, 1, 2, 2, 1, 0, 1, 2, 3, 1, 0, 2, 1, 4, 1 Step 1 . List every unique value: 0, 1, 2, 3, 4 . Step 2 . Tally how many times each appears. Step 3 . Check the total: 4 + 8 + 5 + 2 + 1 = 20 ✓

Pets owned — frequency table
Pets owned — frequency table

Example 1 — Worked example — cricket shots (categories)

A batter faces 17 balls. The shot types are: Drive, Cut, Drive, Pull, Cut, Sweep, Drive, Cut, Pull, Drive, Block, Cut, Drive, Sweep, Cut, Drive, Pull Step 1 . List every category: Drive, Cut, Pull, Sweep, Block. Step 2 . Tally: Shot Frequency Drive 6 Cut 5 Pull 3 Sweep 2 Block 1 Total 17 Check: 6 + 5 + 3 + 2 + 1 = 17 ✓ The mode (most common shot) is Drive . For labelled categories like this you usually stop at the frequency table and the mode — mean and median of the labels do not make sense.

Shot	Frequency
Drive	$6$
Cut	$5$
Pull	$3$
Sweep	$2$
Block	$1$
Total	$17$

Practice 1 . Build a frequency table for: 2, 0, 1, 2, 1, 3, 0, 1, 1, 2, 0, 2, 1, 3, 1, 0, 1, 2 Answer: Value Frequency 04172532 Total 18 Check: 4 + 7 + 5 + 2 = 18 ✓

Value	Frequency
$0$	$4$
$1$	$7$
$2$	$5$
$3$	$2$
Total	$18$

Practice 2 — more cricket shots. Build a frequency table for these 20 shots: Cut, Drive, Cut, Pull, Cut, Drive, Sweep, Cut, Drive, Cut, Pull, Cut, Drive, Hook, Cut, Drive, Cut, Sweep, Cut, Drive Answer: Shot Frequency Cut 9 Drive 6 Pull 2 Sweep 2 Hook 1 Total 20 Check: 9 + 6 + 2 + 2 + 1 = 20 ✓. Mode = Cut .

Shot	Frequency
Cut	$9$
Drive	$6$
Pull	$2$
Sweep	$2$
Hook	$1$
Total	$20$

E4 — Mean, Median, and Mode from a Frequency Table

You cannot "eyeball" the middle of a frequency table. Use these rules: Weighted mean weighted mean = \frac{\sum ( value \times frequency )}{total frequency} Median — make a cumulative frequency column, then walk down until you pass the middle position(s). Mode — the value with the highest frequency . Key idea: for even n, the two middle positions both matter; for odd n, one middle position is enough.

Example 1 — Worked example — using the pets table

Using the pets frequency table (total n = 20). Mean (weighted): \frac{0 \times 4 + 1 \times 8 + 2 \times 5 + 3 \times 2 + 4 \times 1}{20} = \frac{28}{20} = 1.4 Cumulative frequency: Pets Frequency Cumulative frequency 0441812251732194120 Middle positions for n = 20 are 10 and 11 . Both fall in the " 1 pet" row (cumulative reaches 12 there). Median = 1 Mode = 1 (frequency 8 is the largest). Answers: mean 1.4, median 1, mode 1 .

Pets	Frequency	Cumulative frequency
$0$	$4$	$4$
$1$	$8$	$12$
$2$	$5$	$17$
$3$	$2$	$19$
$4$	$1$	$20$

Practice 1 . Books borrowed in a week: Books Frequency 0315263442 Find mean, median, and mode. (n = 20) Answers: Mean = \frac{0 \times 3 + 1 \times 5 + 2 \times 6 + 3 \times 4 + 4 \times 2}{20} = \frac{37}{20} = 1.85 . Cumulative: 3, 8, 14, 18, 20 \to positions 10 and 11 are in the " 2 books" row \to median 2 . Mode 2 (frequency 6).

Books	Frequency
$0$	$3$
$1$	$5$
$2$	$6$
$3$	$4$
$4$	$2$

Practice 2 . Minutes of phone use (grouped as whole minutes for this table): Minutes Frequency 1224374552 Find mean, median, and mode. (n = 20) Answers: Mean = \frac{1 \times 2 + 2 \times 4 + 3 \times 7 + 4 \times 5 + 5 \times 2}{20} = \frac{61}{20} = 3.05 . Cumulative: 2, 6, 13, 18, 20 \to positions 10 and 11 are in the " 3 " row \to median 3 . Mode 3 (frequency 7).

Minutes	Frequency
$1$	$2$
$2$	$4$
$3$	$7$
$4$	$5$
$5$	$2$

E5 — Visualising Frequency Data with a Bar Chart

A bar chart shows each category as a separate bar. For discrete counts (0 pets, 1 pet, 2 pets, \dots) the bars should not touch — these are separate categories, not a continuous scale. Use a bar chart here, not a line graph, because the horizontal values are discrete categories. Key idea: discrete categories \to bar chart with gaps between bars.

Bar chart: number of pets owned (0–4) versus number of students, peaking at 1 pet with 8 students. — $Pets owned by 20 students — bars do not touch because the counts are discrete.$

Example 1 — Worked example — reading the pets chart

From the bar chart (and the table behind it): The tallest bar is 1 pet (frequency 8) — that matches the mode. Bars for 0, 1, 2, 3, 4 stand separately with gaps — correct for discrete data. Heights add to 4 + 8 + 5 + 2 + 1 = 20 — same total as the class size.

Bar chart of cricket shot types: Drive 6, Cut 5, Pull 3, Sweep 2, Block 1. — $Same idea with labelled categories — Drive, Cut, Pull, Sweep, Block — bars still have gaps.$

Example 1 — Worked example — reading the cricket shots chart

From the cricket shots bar chart (17 balls): Tallest bar = Drive (6) \to mode. Cut is next (5) — almost as common as the drive. Heights: 6 + 5 + 3 + 2 + 1 = 17 ✓ Categories are words, not numbers — still a bar chart with gaps, not a line graph.

Practice 1 . Sketch (or complete) a bar chart for this frequency table of siblings in a class of 18 : Siblings Frequency 04172532 Label the axes "Number of siblings" and "Frequency", leave gaps between bars, and give the chart a title.

Siblings	Frequency
$0$	$4$
$1$	$7$
$2$	$5$
$3$	$2$

Bar chart: number of siblings (0–3) versus frequency for 18 students, peaking at 1 sibling with 7 students. — $Model answer for Practice 1 — gaps between bars, axes labelled, title included.$

Practice . Using the cricket shots chart above, answer:1Which shot is the mode?2How many more cuts than pulls were there?3What fraction of the balls were drives?Answers: . Drive. . . . .

More lessons in Descriptive Statistics · Next: Types of Data · Previous: Introduction

E1 — What Is a Statistic?

A statistic is a single number that summarises a set of data. Four of the most useful ones answer four different questions: Mean — what is a "typical" value? Add all the values, then divide by how many there are. Median — what is the middle value once the list is sorted? Mode — which value appears most often? Range — how spread out is the data? Highest value minus lowest value. Key idea: mean / median / mode describe the centre; range describes the spread.

E2 — Mean, Median, Mode, and Range from a List

When you have a raw list of numbers, follow this order every time: 1 Sort the data (needed for the median; also makes the mode easier to spot). 2 Mean = sum \div count. 3 Median = the middle value. With an odd count, take the centre position. With an even count, average the two middle values. 4 Mode = the value with the highest frequency (a list can have no mode, one mode, or several modes if there is a tie). 5 Range = largest - smallest. Key idea: sort first — then mean, median, mode, and range are all straightforward.

Example 1 — Worked example — nine test scores

Scores (out of 100): 72, 85, 91, 68, 85, 77, 90, 63, 79 Step 1 — sort. 63, 68, 72, 77, 79, 85, 85, 90, 91 Step 2 — mean. Sum = 710, count = 9 : Mean = \frac{710}{9} = 78.9 (1 d.p.) Step 3 — median. Nine values \to middle position is the 5th. Median = 79 (If there had been an even count, you would average the two middle values.) Step 4 — mode. 85 appears twice; every other value appears once. Mode = 85 Step 5 — range. Range = 91 - 63 = 28 Answers: mean 78.9, median 79, mode 85, range 28 .

Practice 1 . For the data 54, 61, 58, 61, 70, 65, 49, find the mean (1 d. p .), median, mode, and range. Answers: sorted 49, 54, 58, 61, 61, 65, 70 . Mean = \frac{418}{7} = 59.7 . Median = 61 . Mode = 61 . Range = 70 - 49 = 21 .

Practice 2 . For the data 88, 76, 92, 76, 81, 85, 76, 90, find the mean, median, mode, and range. Answers: sorted 76, 76, 76, 81, 85, 88, 90, 92 . Mean = \frac{664}{8} = 83 . Median = \frac{81 + 85}{2} = 83 (even count). Mode = 76$ . R an g e = 92 - 76 = 16$.

Practice 3 — cricket runs. Runs scored off 12 balls: 0, 4, 1, 6, 2, 4, 0, 1, 4, 6, 2, 4 . Find the mean (1 d. p .), median, mode, and range. Answers: sorted 0, 0, 1, 1, 2, 2, 4, 4, 4, 4, 6, 6 . Mean = \frac{34}{12} = 2.8 . Median = \frac{2 + 4}{2} = 3 . Mode = 4 . Range = 6 - 0 = 6 .

E3 — Organising Data into a Frequency Table

A frequency table lists each unique value (or category) once and shows how many times it occurs. That works for numbers (0 pets, 1 pet, \dots) and for labels (Drive, Cut, Pull, \dots). The tally method is the same. Key idea: after you fill the table, add the frequencies — they must equal the original total count. That check catches most tally mistakes.

Example 1 — Worked example — pets owned (20 students)

Raw data: 0, 1, 2, 1, 0, 3, 1, 2, 2, 1, 0, 1, 2, 3, 1, 0, 2, 1, 4, 1 Step 1 . List every unique value: 0, 1, 2, 3, 4 . Step 2 . Tally how many times each appears. Step 3 . Check the total: 4 + 8 + 5 + 2 + 1 = 20 ✓

Pets owned — frequency table
Pets owned — frequency table

Example 1 — Worked example — cricket shots (categories)

A batter faces 17 balls. The shot types are: Drive, Cut, Drive, Pull, Cut, Sweep, Drive, Cut, Pull, Drive, Block, Cut, Drive, Sweep, Cut, Drive, Pull Step 1 . List every category: Drive, Cut, Pull, Sweep, Block. Step 2 . Tally: Shot Frequency Drive 6 Cut 5 Pull 3 Sweep 2 Block 1 Total 17 Check: 6 + 5 + 3 + 2 + 1 = 17 ✓ The mode (most common shot) is Drive . For labelled categories like this you usually stop at the frequency table and the mode — mean and median of the labels do not make sense.

Shot	Frequency
Drive	$6$
Cut	$5$
Pull	$3$
Sweep	$2$
Block	$1$
Total	$17$

Practice 1 . Build a frequency table for: 2, 0, 1, 2, 1, 3, 0, 1, 1, 2, 0, 2, 1, 3, 1, 0, 1, 2 Answer: Value Frequency 04172532 Total 18 Check: 4 + 7 + 5 + 2 = 18 ✓

Value	Frequency
$0$	$4$
$1$	$7$
$2$	$5$
$3$	$2$
Total	$18$

Practice 2 — more cricket shots. Build a frequency table for these 20 shots: Cut, Drive, Cut, Pull, Cut, Drive, Sweep, Cut, Drive, Cut, Pull, Cut, Drive, Hook, Cut, Drive, Cut, Sweep, Cut, Drive Answer: Shot Frequency Cut 9 Drive 6 Pull 2 Sweep 2 Hook 1 Total 20 Check: 9 + 6 + 2 + 2 + 1 = 20 ✓. Mode = Cut .

Shot	Frequency
Cut	$9$
Drive	$6$
Pull	$2$
Sweep	$2$
Hook	$1$
Total	$20$

E4 — Mean, Median, and Mode from a Frequency Table

You cannot "eyeball" the middle of a frequency table. Use these rules: Weighted mean weighted mean = \frac{\sum ( value \times frequency )}{total frequency} Median — make a cumulative frequency column, then walk down until you pass the middle position(s). Mode — the value with the highest frequency . Key idea: for even n, the two middle positions both matter; for odd n, one middle position is enough.

Example 1 — Worked example — using the pets table

Using the pets frequency table (total n = 20). Mean (weighted): \frac{0 \times 4 + 1 \times 8 + 2 \times 5 + 3 \times 2 + 4 \times 1}{20} = \frac{28}{20} = 1.4 Cumulative frequency: Pets Frequency Cumulative frequency 0441812251732194120 Middle positions for n = 20 are 10 and 11 . Both fall in the " 1 pet" row (cumulative reaches 12 there). Median = 1 Mode = 1 (frequency 8 is the largest). Answers: mean 1.4, median 1, mode 1 .

Pets	Frequency	Cumulative frequency
$0$	$4$	$4$
$1$	$8$	$12$
$2$	$5$	$17$
$3$	$2$	$19$
$4$	$1$	$20$

Practice 1 . Books borrowed in a week: Books Frequency 0315263442 Find mean, median, and mode. (n = 20) Answers: Mean = \frac{0 \times 3 + 1 \times 5 + 2 \times 6 + 3 \times 4 + 4 \times 2}{20} = \frac{37}{20} = 1.85 . Cumulative: 3, 8, 14, 18, 20 \to positions 10 and 11 are in the " 2 books" row \to median 2 . Mode 2 (frequency 6).

Books	Frequency
$0$	$3$
$1$	$5$
$2$	$6$
$3$	$4$
$4$	$2$

Practice 2 . Minutes of phone use (grouped as whole minutes for this table): Minutes Frequency 1224374552 Find mean, median, and mode. (n = 20) Answers: Mean = \frac{1 \times 2 + 2 \times 4 + 3 \times 7 + 4 \times 5 + 5 \times 2}{20} = \frac{61}{20} = 3.05 . Cumulative: 2, 6, 13, 18, 20 \to positions 10 and 11 are in the " 3 " row \to median 3 . Mode 3 (frequency 7).

Minutes	Frequency
$1$	$2$
$2$	$4$
$3$	$7$
$4$	$5$
$5$	$2$

E5 — Visualising Frequency Data with a Bar Chart

A bar chart shows each category as a separate bar. For discrete counts (0 pets, 1 pet, 2 pets, \dots) the bars should not touch — these are separate categories, not a continuous scale. Use a bar chart here, not a line graph, because the horizontal values are discrete categories. Key idea: discrete categories \to bar chart with gaps between bars.

Example 1 — Worked example — reading the pets chart

From the bar chart (and the table behind it): The tallest bar is 1 pet (frequency 8) — that matches the mode. Bars for 0, 1, 2, 3, 4 stand separately with gaps — correct for discrete data. Heights add to 4 + 8 + 5 + 2 + 1 = 20 — same total as the class size.

Example 1 — Worked example — reading the cricket shots chart

From the cricket shots bar chart (17 balls): Tallest bar = Drive (6) \to mode. Cut is next (5) — almost as common as the drive. Heights: 6 + 5 + 3 + 2 + 1 = 17 ✓ Categories are words, not numbers — still a bar chart with gaps, not a line graph.

Practice 1 . Sketch (or complete) a bar chart for this frequency table of siblings in a class of 18 : Siblings Frequency 04172532 Label the axes "Number of siblings" and "Frequency", leave gaps between bars, and give the chart a title.

Siblings	Frequency
$0$	$4$
$1$	$7$
$2$	$5$
$3$	$2$

Practice . Using the cricket shots chart above, answer:1Which shot is the mode?2How many more cuts than pulls were there?3What fraction of the balls were drives?Answers: . Drive. . . . .