Plot the Dots

The first thing you should do whenever you find yourself with data, is to visualise the data - typically to plot it on a graph.
Which graph you use will depend on what data you have.
For example, suppose you have
(NOTE EACH OF THESE SECTIONS IS EXPLAINED IN MORE DETAIL ON THE LINKS)

Scores, or Numbers against departments
For example, this could include:

numbers of service user contacts per person (NOTE if you have numbers of contacts per week for a number of weeks, see below)
patient satisfaction scores for each department or organisation
smoking quitters per 1,000 population by GP practice
average length of stay (LOS) by speciality and by ward
income (in HRG) per consultant

Plot
a histograph or histogram
If it's a normal curve, you can boxplot the averages and if there's something
to go on then compare averages with
Student's t test for two averages
ANOVA analysis of variables for more than two averages

For a two-way analysis (where you have two comparators eg sex, hospital or speciality, ward)

use ANOVA analysis of variables for two or more ways
use Chi-squared () to test whether the observations are significantly different from what you would 'expect'

Change over time
For example:

waiting time before a major change in procedure such as service redesign, and after, with scores collected weekly over a 10 month period
max heart rate during exercise (per session) for a patient going through pulmonary rehab over a 14 week period
travel time to work before, during and after a road widening scheme

Draw a Run Chart
If it shows a change, then you can consider comparing averages using typically Student's t test for the average before and the average after

Compare pairs of values
For example:

BMIs (Body Mass Indexs) vs Blood Pressures of a whole lot of people
Cost vs Tariff price
Patient Survey satisfaction score vs Practice List size for GP practices

Plot a scatter graph
if you can see a pattern emerging in the scatter graph, then

calculate the Correlation Coefficient and regression line
possibly plan to do root cause analysis or design an experiment to determine if there's a causative relationship between the values

Plot a bubble graph or Boston Matrix
Only use these fancy illustrations if you have particular kinds of data,
for example Opportunity/ Cost for a series of projects. Bubble graphs can be used to show a third dimension of data (eg adding the waist circumference to the BMI vs Blood Pressure graph) but generally makes it all look very complicated with more than about 10 data points.