What are my standings toward NPCs and what do they do?
The standings set by the game are indicators of your stature towards NPC entities: agents, corporations, and factions. All NPC corporations also have standings between themselves as well as with player corporations.
All agents have six standings relationships with a capsuleer, which can be viewed by getting "info" on the agent in game, and going to the standings tab. The agent, her corporation and her faction have derived standings with the pilot and his corporation. All level 1 agents (except for storyline agents) should be available to every pilot. All level 2 thru level 5 agents require a minimum derived standing, which is taken as the highest of the "Standings With You" on the agent's Standings Tab (in-game).
The equation is
where E is your effective standing, B is your base standing, and S is your Diplomacy skill level (if B <0) or Connections skill level (otherwise).
So, the Diplomacy skill works in the following way for someone with a -1.0 standing from a NPC corp/faction:
To calculate how high your base standing needs to be in order to reach a certain effective standing, the formula can be rearranged to
How do I gain, or lose standing toward NPCs?
The only way to increase your standing towards an agent or a corporation is by successfully completing missions for that agent/corp. To increase your standing towards a faction you need to successfully complete storyline missions for that faction or one friendly to it.
You can lose standing towards an agent or corporation by failing a mission, or by declining two missions from the same agent in a 4 hours interval. Similarly you lose standing towards a faction is you fail storyline missions of that faction or one friendly to it, but also if you successfully complete a storyline mission for an enemy faction or if you destroy ships or structures belonging to that particular faction.
Increase standing changes are derived as a percentage change of the difference between current "true" standing (unmodified) and 10.
For instance a 3% increase would work out the following way in each situation:
Decrease standing changes work the same way. Except the endpoint is -10.
For instance a 3% decrease would work out the following way in each situation:
Completing a storyline mission will have an impact on all your faction standings via "derived standings" as stated in your faction standing transaction window (available by right clicking any standing entries in your character sheet and selecting "Show Transactions"). Those derived standings depends on how other factions perceive the faction you completed such important missions for. There is also a limit on how those derived standings can increase or decrease your own standings, as those factions will never hate or like you more than the faction you worked for. 
See the storyline mission page for more details on how those derived standings are calculated.
NPC Faction standings towards Player corporations
NPC faction to player corporation standings are important to join factional warfare militias.
NPC factions/corps standings toward a player corporation are calculated in this way:
You take the standing (without skills included) of each corporation member (towards the NPC entity in question) on an active subscribing account, add them all up and then divide by the number of members that have the standing already in their character sheet. In rare circumstances characters can have standing towards an entity that is not listed in their character sheet. Check by looking up the entity and then looking at the "Standings" tab. The ones who do not have a standing towards the entity are not taken into account. The standings will update to the average after every downtime.
It is also worth noting that members standings towards NPC corps do not begin to affect their global corp standing towards the relevant NPC corp until they have been a member of their current corp for at least 7 days.
Standings between player entities
Player characters and corporations can also have standings from and towards each other, which can be adjusted by the players or corporations directly. In addition to that player alliances can adjust standings towards each other and to player corporations.
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